Homothetic center - Knowledge

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of the three circles (Figure 9). Offset each center point perpendicularly to the plane by a distance equal to the corresponding radius. The centers can be offset to either side of the plane. The three offset points define a single plane. In that plane we build three lines through each pair of points.

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identical circles. In the limit of two circles with the same radius but distinct centers moving to having the same center, the external center is the point at infinity corresponding to the slope of the line of centers, which can be anything, so no limit exists for all possible pairs of such circles.

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If the two circles are identical (same center, same radius), the internal center is their common center, but there is no well-defined external center – properly, the function from the parameter space of two circles in the plane to the external center has a non-removable discontinuity on the locus of

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three given circles (Figure 11). By conjugate we imply that both tangent circles belong to the same family with respect to any one of the given pairs of circles. As we've already seen, the radical axis of any two tangent circles from the same family passes through the homothetic center of the two

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If the circles have the same center but different radii, both the external and internal coincide with the common center of the circles. This can be seen from the analytic formula, and is also the limit of the two homothetic centers as the centers of the two circles are varied until they coincide,

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More generally, taking both radii with the same sign (both positive or both negative) yields the inner center, while taking the radii with opposite signs (one positive and the other negative) yields the outer center. Note that the equation for the inner center is valid for any values (unless both

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relative to the circles are equal. The radical axis is always perpendicular to the line of centers, and if two circles intersect, their radical axis is the line joining their points of intersection. For three circles, three radical axes can be defined, one for each pair of circles

3255:. Given the three circles, the homothetic centers can be found and thus the radical axis of a pair of solution circles. Of course, there are infinitely many circles with the same radical axis, so additional work is done to find out exactly which two circles are the solution. 2778: 3171:

Repeating the above procedure for different combinations of homothetic centers (in our method this is determined by the side to which we offset the centers of the circles) would yield a total of four lines — three homothetic centers on each line (Figure 10).

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Any pair of circles has two centers of similarity, therefore, three circles would have six centers of similarity, two for each distinct pair of given circles. Remarkably, these six points lie on four lines, three points on each line. Here is one way to show

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If we construct the tangent circles for every possible pair of antihomologous points we get two families of circles - one for each homothetic center. The family of circles of the external homothetic center is such that every tangent circle either contains

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Circles are geometrically similar to one another and mirror symmetric. Hence, a pair of circles has both types of homothetic centers, internal and external, unless the centers are equal or the radii are equal; these exceptional cases are treated after

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Conversely, if both radii are zero (two points) but the points are distinct, the external center can be defined as the point at infinity corresponding to the slope of the line of centers, but there is no well-defined internal center.

1875:{\displaystyle {\begin{aligned}&\triangle PIR\cong \triangle P'\!IR'\\&\implies \angle RPI=\angle IP'\!R'=\alpha \\&\implies \angle RS'\!Q'=\angle PP'\!R'=\alpha \quad {\text{(inscribed angle theorem)}}\end{aligned}}} 181:

to one another; in other words they must have the same angles at corresponding points and differ only in their relative scaling. The homothetic center and the two figures need not lie in the same plane; they can be related by a

193:. A clockwise angle in one figure would correspond to a counterclockwise angle in the other. Conversely, if the center is external, the two figures are directly similar to one another; their angles have the same sense. 1018: 1289: 2658: 2387: 1216: 913:

to be the point at infinity corresponding to the slope of this line. This is also the limit of the external center if the centers of the circles are fixed and the radii are varied until they are equal.

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of the centers of the circles, weighted by the opposite circle's radius – distance from center of circle to inner center is proportional to that radius, so weighting is proportional to the

2944:{\displaystyle {\frac {\overline {EP}}{\overline {EP'}}}={\frac {\overline {EQ}}{\overline {EQ'}}};\quad {\overline {EP}}\cdot {\overline {EQ'}}={\overline {EQ}}\cdot {\overline {EP'}}.} 189:

Homothetic centers may be external or internal. If the center is internal, the two geometric figures are scaled mirror images of one another; in technical language, they have opposite

956:. These four points lie on a circle that intersects the two given circles; the lines through the intersection points of the new circle with the two given circles must intersect at the 2010: 1907: 1587: 1484: 3034: 850:

to both circles (a bitangent line) passes through one of the homothetic centers, as it forms right angles with both the corresponding diameters, which are thus parallel; see

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If one radius is zero but the other is non-zero (a point and a circle), both the external and internal center coincide with the point (center of the radius zero circle).

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radii zero or one is the negative of the other), but the equation for the external center requires that the radii be different, otherwise it involves division by zero.

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of points which are common to two distinct and non-parallel planes is a line then necessarily these three points lie on such line. From the similarity of triangles

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For each pair of antihomologous points of two circles exists a third circle which is tangent to the given ones and touches them at the antihomologous points.

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Any two pairs of antihomologous points can be used to find a point on the radical axis. Consider the two rays emanating from the external homothetic center

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are parallel to the line of centers (both for secant lines and the bitangent lines) and thus have no intersection. An external center can be defined in the

210:(Figure 3). Circles with radius zero can also be included (see exceptional cases), and negative radius can also be used, switching external and internal. 3228:

given circles. Since the tangent circles are common for all three pairs of given circles then their homothetic centers all belong to the radical axis of

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The external center can be computed by the same equation, but considering one of the radii as negative; either one yields the same equation, which is:

3199:(pink). Each pair of given circles has a homothetic center which belongs to the radical axis of the two tangent circles. Since the radical axis is a 869:

in the figure above. Conversely, if the circles fall on the same side of the line, it passes through the external homothetic center (not pictured).

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given circles or none (Figure 6). On the other hand, the circles from the other family always contain only one of the given circles (Figure 7).

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Figure 4: Lines through corresponding antihomologous points intersect on the radical axis of the two given circles (green and blue). The points

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In general, a line passing through a homothetic center intersects each of its circles in two places. Of these four points, two are said to be

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is an external homothetic center for the two triangles. The size of each figure is proportional to its distance from the homothetic center.

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holding the radii equal. There is no line of centers, however, and the synthetic construction fails as the two parallel lines coincide.

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drawn through corresponding endpoints of those radii, which are homologous points, intersect each other and the line of centers at the

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The opposite is also true — every circle which is tangent to two other circles touches them at a pair of antihomologous points.

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in Figure 4. These rays intersect the two given circles (green and blue in Figure 4) in two pairs of antihomologous points,

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drawn through one endpoint and the opposite endpoint of its counterpart intersects each other and the line of centers at the

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To show this, consider two rays from the homothetic center, intersecting the given circles (Figure 8). Two tangent circles

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for the second ray. These four points lie on a single circle, that intersects both given circles. By definition, the line

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If the two tangent circle touch collinear pairs of antihomologous point — as in Figure 5 — then because of the homothety

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When two rays from the same homothetic center intersect the circles, each set of antihomologous points lie on a circle.

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for details. If the circles fall on opposite sides of the line, it passes through the internal homothetic center, as in

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Figure 9: In a three circle configuration, three homothetic centers (one for each pair of circles) lie on a single line

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exist which touch the given circles at the antihomologous points. As we've already shown these points lie on a circle

2559:{\displaystyle \angle O_{1}PQ=\angle O_{1}QP=\angle O_{2}P'\!Q'=\angle O_{2}Q'\!P'=\angle T_{1}QP'=\angle T_{1}P'\!Q.} 134:, the two figures are directly similar to one another; their angles have the same rotational sense. If the center is 79: 57: 50: 760:{\displaystyle (x_{e},y_{e})={\frac {-r_{2}}{r_{1}-r_{2}}}(x_{1},y_{1})+{\frac {r_{1}}{r_{1}-r_{2}}}(x_{2},y_{2}).} 564:{\displaystyle (x_{0},y_{0})={\frac {r_{2}}{r_{1}+r_{2}}}(x_{1},y_{1})+{\frac {r_{1}}{r_{1}+r_{2}}}(x_{2},y_{2}).} 206:. These two homothetic centers lie on the line joining the centers of the two given circles, which is called the 173:

The external (above) and internal (below) homothetic centers of the two circles (red) are shown as black points.

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Figure 5: Every circle which is tangent to two given circles touches them at a pair of antihomologous points

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If the circles have the same radius (but different centers), they have no external homothetic center in the

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All circles from a tangent family have a common radical center and it coincides with the homothetic center.

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For a given pair of circles, the internal and external homothetic centers may be found in various ways. In

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is the radical axis of the new circle with the blue given circle. These two lines intersect at the point

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if radii drawn to them make the same angle with the line connecting the centers; for example, the points

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Advanced Euclidean Geometry: An Elementary treatise on the geometry of the Triangle and the Circle

944: 3284: 2197:, which is the radical center of the new circle and the two given circles. Therefore, the point 2167: 1219: 957: 190: 138:, the two figures are scaled mirror images of one another; their angles have the opposite sense. 3179: 2099:{\displaystyle {\overline {IQ}}\cdot {\overline {IP'}}={\overline {IS}}\cdot {\overline {IR'}}.} 1996:{\displaystyle {\overline {IP}}\cdot {\overline {IQ'}}={\overline {IR}}\cdot {\overline {IS'}}.} 1676:{\displaystyle {\overline {EP}}\cdot {\overline {EQ'}}={\overline {ER}}\cdot {\overline {ES'}}.} 1573:{\displaystyle {\overline {EQ}}\cdot {\overline {EP'}}={\overline {ES}}\cdot {\overline {ER'}}.} 3289: 881:: in analytic geometry this results in division by zero, while in synthetic geometry the lines 61: 2752:. Then the intersecting point of the two radical axes must also belong to the radical axis of 3270: 3252: 3167:

Figure 10: All six homothetic centers (dots) of three circles lie on four lines (thick lines)

3120:{\displaystyle {\frac {\overline {H\!_{AB}B}}{\overline {H\!_{AB}A}}}={\frac {r_{B}}{r_{A}}}} 178: 123: 2170:. Tangents drawn from the radical center to the three circles would all have equal length. 3248: 1474: 1299: 3145:

is in fact the homothetic center of the corresponding two circles. We can do the same for

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in Figure 4. Points which are collinear with respect to the homothetic center but are

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is the radical axis of the new circle with the green given circle, whereas the line

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Figure 8: The radical axis of tangent circles passes through the radical center

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BSHM Bulletin: Journal of the British Society for the History of Mathematics

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Point from which two similar geometric figures can be scaled to each other

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Figure 2: Two geometric figures related by an external homothetic center

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Figure 11: The blue line is the radical axis of the two tangent circles

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is their external homothetic center. We construct an arbitrary ray from

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Figure 3: Two circles have both types of homothetic centers, internal (

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Figure 7: Family of tangent circles for the internal homothetic center

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Figure 6: Family of tangent circles for the external homothetic center

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with respect to the two tangent circles are equal which means that

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is the homothetic center. From that similarity, it follows that

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If two geometric figures possess a homothetic center, they are

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is also isosceles and a circle can be constructed with center

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this means that the three homothetic centers are collinear

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This circle is tangent to the two given circles in points

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also lies on the radical axis of the two given circles.

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The proof is similar for the internal homothetic center

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The proof for the other pair of antihomologous points (

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The lines pierce the plane of circles in the points

3119: 2943: 2652: 2558: 2372: 2098: 1995: 1874: 1675: 1572: 1465: 1357: 1283: 1211:{\displaystyle \angle ESQ=\angle ES'\!Q'=\alpha .} 1210: 1142: 759: 563: 3342:"The tangency problem of Apollonius: three looks" 3070: 3046: 2549: 2489: 2456: 2108: 1847: 1821: 1780: 1727: 1413: 1358:{\displaystyle \angle QSR'=180^{\circ }-\alpha .} 1269: 1243: 1190: 1048: 3377: 846:As a limiting case of this construction, a line 213: 1284:{\displaystyle \angle EP'\!R'=\angle ES'\!Q'.} 995:Pairs of antihomologous points lie on a circle 298:radius. Denoting the centers of the circles 2728:and thus the two rays are radical axes for 3330: 3223:be a conjugate pair of circles tangent to 1806: 1802: 1750: 1746: 811:homothetic center. Conversely, the lines 250:) from each homothetic center. The points 160:are both clockwise and equal in magnitude. 2205:Tangent circles and antihomologous points 80:Learn how and when to remove this message 3178: 3162: 2977: 2697: 2681: 2673: 2213: 963:of the three circles, which lies on the 943: 290:, the internal homothetic center is the 217: 168: 140: 93: 43:This article includes a list of general 3138:are the radii of the circles) and thus 14: 3378: 3339: 3175:Here is yet another way to prove this. 1580:In the same way, it can be shown that 3309: 2285:. It is easily proven that triangles 122:) is a point from which at least two 2246:which intersects the two circles in 933:Homologous and antihomologous points 779:with the line of centers. The lines 246:) are proportional to the distance ( 29: 2963:Homothetic centers of three circles 939:Homologous points (computer vision) 164: 24: 2528: 2501: 2468: 2435: 2413: 2391: 1833: 1807: 1766: 1751: 1716: 1701: 1584:can be inscribed in a circle and 1399: 1379: 1316: 1255: 1229: 1176: 1161: 1037: 1022: 266:are homologous, as are the points 49:it lacks sufficient corresponding 25: 3407: 3244:e.g., they lie on a single line. 2222:Let our two circles have centers 2007:can be inscribed in a circle and 130:of one another. If the center is 872: 34: 3335:. New York: Dover Publications. 2858: 1900:lie on a circle. Then from the 1888:is seen in the same angle from 1862: 1063: 3303: 3247:This property is exploited in 2109:Relation with the radical axis 1803: 1747: 751: 725: 680: 654: 604: 578: 555: 529: 484: 458: 413: 387: 13: 1: 3296: 2959:belongs to the radical axis. 230:). The radii of the circles ( 3084: 3060: 2933: 2910: 2892: 2869: 2849: 2831: 2811: 2793: 2642: 2617: 2365: 2340: 2088: 2065: 2047: 2024: 1985: 1962: 1944: 1921: 1665: 1642: 1624: 1601: 1562: 1539: 1521: 1498: 1131: 1113: 1093: 1075: 852:tangent lines to two circles 214:Computing homothetic centers 186:from the homothetic center. 7: 3258: 2308:are similar because of the 1902:intersecting chords theorem 952:are antihomologous, as are 362:and denoting the center by 10: 3412: 2966: 983:homologous are said to be 936: 196: 3361:10.1080/17498430601148911 3276:Homothetic transformation 1865:(inscribed angle theorem) 1015:They are similar because 967:of the two given circles. 126:figures can be seen as a 3320:--A Wolfram Web Resource 2276:until they intersect in 2181:for the first ray, and 1220:inscribed angle theorem 128:dilation or contraction 64:more precise citations. 3249:Joseph Diaz Gergonne's 3204: 3168: 3121: 2983: 2945: 2703: 2687: 2679: 2654: 2560: 2374: 2219: 2100: 1997: 1876: 1677: 1574: 1473:which means it can be 1467: 1359: 1285: 1212: 1144: 987:; for example, points 968: 761: 565: 283: 174: 161: 103: 3340:Kunkel, Paul (2007), 3313:Antihomologous Points 3271:Similarity (geometry) 3182: 3166: 3122: 2981: 2946: 2701: 2685: 2677: 2655: 2561: 2375: 2217: 2101: 1998: 1877: 1678: 1575: 1475:inscribed in a circle 1468: 1360: 1286: 1213: 1145: 947: 762: 566: 221: 172: 144: 124:geometrically similar 97: 3310:Weisstein, Eric W., 3251:general solution to 3035: 2779: 2592: 2388: 2320: 2011: 1908: 1693: 1588: 1485: 1376: 1313: 1226: 1158: 1019: 575: 384: 120:center of similitude 116:center of similarity 98:Figure 1: The point 18:Center of similitude 3331:Johnson RA (1960). 3290:Apollonius' problem 3253:Apollonius' problem 2951:Thus the powers of 2113:Two circles have a 1481:, it follows that 1002:Consider triangles 843:homothetic center. 346:and their radii by 3386:Euclidean geometry 3205: 3169: 3117: 2984: 2941: 2704: 2688: 2680: 2650: 2556: 2370: 2220: 2096: 1993: 1872: 1870: 1673: 1570: 1463: 1355: 1281: 1208: 1140: 969: 773:synthetic geometry 757: 561: 284: 175: 162: 104: 3396:Geometric centers 3266:Intercept theorem 3115: 3088: 3087: 3063: 2936: 2913: 2895: 2872: 2853: 2852: 2834: 2815: 2814: 2796: 2645: 2620: 2368: 2343: 2091: 2068: 2050: 2027: 1988: 1965: 1947: 1924: 1866: 1668: 1645: 1627: 1604: 1565: 1542: 1524: 1501: 1135: 1134: 1116: 1097: 1096: 1078: 723: 652: 527: 456: 288:analytic geometry 112:homothetic center 90: 89: 82: 16:(Redirected from 3403: 3371: 3346: 3336: 3322: 3321: 3307: 3243: 3222: 3198: 3158: 3151: 3144: 3137: 3126: 3124: 3123: 3118: 3116: 3114: 3113: 3104: 3103: 3094: 3089: 3083: 3079: 3078: 3064: 3059: 3055: 3054: 3040: 3039: 3030: 3007: 2958: 2954: 2950: 2948: 2947: 2942: 2937: 2932: 2931: 2919: 2914: 2909: 2901: 2896: 2891: 2890: 2878: 2873: 2868: 2860: 2854: 2848: 2847: 2835: 2830: 2822: 2821: 2816: 2810: 2809: 2797: 2792: 2784: 2783: 2771: 2767: 2751: 2727: 2723: 2670: 2663: 2659: 2657: 2656: 2651: 2646: 2641: 2637: 2636: 2626: 2621: 2616: 2615: 2607: 2606: 2596: 2587: 2578: 2565: 2563: 2562: 2557: 2548: 2540: 2539: 2524: 2513: 2512: 2497: 2488: 2480: 2479: 2464: 2455: 2447: 2446: 2425: 2424: 2403: 2402: 2379: 2377: 2376: 2371: 2369: 2364: 2360: 2359: 2349: 2344: 2339: 2335: 2334: 2324: 2312:. 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3032: 3027: 3019: 3013: 3006: 3002: 2998: 2994: 2975: 2971: 2969:Monge's theorem 2965: 2956: 2952: 2924: 2920: 2918: 2902: 2900: 2883: 2879: 2877: 2861: 2859: 2840: 2836: 2823: 2820: 2802: 2798: 2785: 2782: 2780: 2777: 2776: 2769: 2766: 2759: 2753: 2750: 2739: 2729: 2725: 2722: 2715: 2709: 2707: 2668: 2661: 2632: 2628: 2627: 2625: 2608: 2602: 2598: 2597: 2595: 2593: 2590: 2589: 2586: 2580: 2574: 2567: 2541: 2535: 2531: 2517: 2508: 2504: 2490: 2481: 2475: 2471: 2457: 2448: 2442: 2438: 2420: 2416: 2398: 2394: 2389: 2386: 2385: 2355: 2351: 2350: 2348: 2330: 2326: 2325: 2323: 2321: 2318: 2317: 2303: 2293: 2286: 2283: 2277: 2271: 2261: 2255: 2251: 2247: 2243: 2239: 2236: 2229: 2223: 2210: 2207: 2198: 2194: 2190: 2186: 2182: 2178: 2174: 2164: 2157: 2150: 2143: 2136: 2129: 2123: 2111: 2079: 2075: 2073: 2057: 2055: 2038: 2034: 2032: 2016: 2014: 2012: 2009: 2008: 2004: 1976: 1972: 1970: 1954: 1952: 1935: 1931: 1929: 1913: 1911: 1909: 1906: 1905: 1897: 1893: 1889: 1884: 1883: 1869: 1868: 1863: 1848: 1839: 1822: 1813: 1796: 1795: 1781: 1772: 1740: 1739: 1731: 1719: 1696: 1694: 1691: 1690: 1686: 1656: 1652: 1650: 1634: 1632: 1615: 1611: 1609: 1593: 1591: 1589: 1586: 1585: 1581: 1553: 1549: 1547: 1531: 1529: 1512: 1508: 1506: 1490: 1488: 1486: 1483: 1482: 1454: 1450: 1429: 1425: 1414: 1405: 1388: 1377: 1374: 1373: 1369: 1340: 1336: 1325: 1314: 1311: 1310: 1303: 1292: 1270: 1261: 1244: 1235: 1227: 1224: 1223: 1191: 1182: 1159: 1156: 1155: 1151: 1122: 1118: 1105: 1102: 1084: 1080: 1067: 1064: 1052: 1040: 1020: 1017: 1016: 1014: 1003: 997: 988: 976: 960: 953: 949: 942: 935: 907: 901: 894: 888: 882: 875: 867: 861: 855: 837: 831: 824: 818: 812: 805: 799: 792: 786: 780: 776: 745: 741: 732: 728: 716: 712: 703: 699: 698: 692: 688: 686: 674: 670: 661: 657: 645: 641: 632: 628: 627: 620: 616: 612: 610: 598: 594: 585: 581: 576: 573: 572: 549: 545: 536: 532: 520: 516: 507: 503: 502: 496: 492: 490: 478: 474: 465: 461: 449: 445: 436: 432: 431: 425: 421: 419: 407: 403: 394: 390: 385: 382: 381: 377: 370: 363: 360: 353: 347: 343: 336: 329: 322: 315: 312: 305: 299: 280: 273: 267: 264: 257: 251: 247: 244: 237: 231: 227: 223: 216: 208:line of centers 199: 167: 150: 146: 114:(also called a 99: 86: 75: 69: 66: 56:Please help to 55: 39: 35: 28: 23: 22: 15: 12: 11: 5: 3409: 3399: 3398: 3393: 3388: 3373: 3372: 3337: 3327: 3324: 3323: 3301: 3300: 3298: 3295: 3293: 3292: 3287: 3285:radical center 3278: 3273: 3268: 3262: 3260: 3257: 3240: 3233: 3219: 3212: 3195: 3188: 3155: 3148: 3141: 3134: 3130: 3112: 3108: 3102: 3098: 3092: 3086: 3082: 3077: 3074: 3068: 3062: 3058: 3053: 3050: 3044: 3025: 3017: 3004: 3000: 2996: 2964: 2961: 2940: 2935: 2930: 2927: 2923: 2917: 2912: 2908: 2905: 2899: 2894: 2889: 2886: 2882: 2876: 2871: 2867: 2864: 2857: 2851: 2846: 2843: 2839: 2833: 2829: 2826: 2819: 2813: 2808: 2805: 2801: 2795: 2791: 2788: 2764: 2757: 2748: 2737: 2720: 2713: 2649: 2644: 2640: 2635: 2631: 2624: 2619: 2614: 2611: 2605: 2601: 2584: 2572: 2555: 2552: 2547: 2544: 2538: 2534: 2530: 2527: 2523: 2520: 2516: 2511: 2507: 2503: 2500: 2496: 2493: 2487: 2484: 2478: 2474: 2470: 2467: 2463: 2460: 2454: 2451: 2445: 2441: 2437: 2434: 2431: 2428: 2423: 2419: 2415: 2412: 2409: 2406: 2401: 2397: 2393: 2367: 2363: 2358: 2354: 2347: 2342: 2338: 2333: 2329: 2301: 2291: 2281: 2269: 2259: 2234: 2227: 2206: 2203: 2168:radical center 2162: 2155: 2148: 2141: 2134: 2127: 2110: 2107: 2095: 2090: 2085: 2082: 2078: 2072: 2067: 2063: 2060: 2054: 2049: 2044: 2041: 2037: 2031: 2026: 2022: 2019: 1992: 1987: 1982: 1979: 1975: 1969: 1964: 1960: 1957: 1951: 1946: 1941: 1938: 1934: 1928: 1923: 1919: 1916: 1896:, which means 1861: 1858: 1854: 1851: 1845: 1842: 1838: 1835: 1832: 1828: 1825: 1819: 1816: 1812: 1809: 1805: 1801: 1799: 1797: 1794: 1791: 1787: 1784: 1778: 1775: 1771: 1768: 1765: 1762: 1759: 1756: 1753: 1749: 1745: 1743: 1741: 1737: 1734: 1730: 1725: 1722: 1718: 1715: 1712: 1709: 1706: 1703: 1700: 1698: 1672: 1667: 1662: 1659: 1655: 1649: 1644: 1640: 1637: 1631: 1626: 1621: 1618: 1614: 1608: 1603: 1599: 1596: 1569: 1564: 1559: 1556: 1552: 1546: 1541: 1537: 1534: 1528: 1523: 1518: 1515: 1511: 1505: 1500: 1496: 1493: 1479:secant theorem 1462: 1457: 1453: 1449: 1446: 1443: 1440: 1437: 1432: 1428: 1424: 1420: 1417: 1411: 1408: 1404: 1401: 1398: 1394: 1391: 1387: 1384: 1381: 1354: 1351: 1348: 1343: 1339: 1335: 1331: 1328: 1324: 1321: 1318: 1280: 1276: 1273: 1267: 1264: 1260: 1257: 1254: 1250: 1247: 1241: 1238: 1234: 1231: 1207: 1204: 1201: 1197: 1194: 1188: 1185: 1181: 1178: 1175: 1172: 1169: 1166: 1163: 1139: 1133: 1128: 1125: 1121: 1115: 1111: 1108: 1101: 1095: 1090: 1087: 1083: 1077: 1073: 1070: 1062: 1058: 1055: 1051: 1046: 1043: 1039: 1036: 1033: 1030: 1027: 1024: 996: 993: 985:antihomologous 958:radical center 934: 931: 905: 899: 892: 886: 874: 871: 865: 859: 835: 829: 822: 816: 803: 797: 790: 784: 756: 753: 748: 744: 740: 735: 731: 727: 719: 715: 711: 706: 702: 695: 691: 685: 682: 677: 673: 669: 664: 660: 656: 648: 644: 640: 635: 631: 623: 619: 615: 609: 606: 601: 597: 593: 588: 584: 580: 560: 557: 552: 548: 544: 539: 535: 531: 523: 519: 515: 510: 506: 499: 495: 489: 486: 481: 477: 473: 468: 464: 460: 452: 448: 444: 439: 435: 428: 424: 418: 415: 410: 406: 402: 397: 393: 389: 375: 368: 358: 351: 341: 334: 327: 320: 310: 303: 278: 271: 262: 255: 242: 235: 215: 212: 198: 195: 166: 163: 88: 87: 42: 40: 33: 26: 9: 6: 4: 3: 2: 3408: 3397: 3394: 3392: 3389: 3387: 3384: 3383: 3381: 3370: 3366: 3362: 3358: 3354: 3350: 3343: 3338: 3334: 3329: 3328: 3319: 3315: 3314: 3306: 3302: 3291: 3288: 3286: 3282: 3279: 3277: 3274: 3272: 3269: 3267: 3264: 3263: 3256: 3254: 3250: 3245: 3239: 3232: 3226: 3218: 3211: 3202: 3194: 3187: 3181: 3177: 3173: 3165: 3161: 3110: 3106: 3100: 3096: 3090: 3080: 3075: 3072: 3066: 3056: 3051: 3048: 3042: 3031:we see that 3029: 3021: 3011: 2991: 2990: 2986:Consider the 2980: 2976: 2970: 2960: 2938: 2928: 2925: 2921: 2915: 2906: 2903: 2897: 2887: 2884: 2880: 2874: 2865: 2862: 2855: 2844: 2841: 2837: 2827: 2824: 2817: 2806: 2803: 2799: 2789: 2786: 2773: 2763: 2756: 2747: 2743: 2736: 2732: 2719: 2712: 2700: 2696: 2694: 2684: 2676: 2672: 2665: 2647: 2638: 2633: 2629: 2622: 2612: 2609: 2603: 2599: 2583: 2577: 2571: 2553: 2550: 2545: 2542: 2536: 2532: 2525: 2521: 2518: 2514: 2509: 2505: 2498: 2494: 2491: 2485: 2482: 2476: 2472: 2465: 2461: 2458: 2452: 2449: 2443: 2439: 2432: 2429: 2426: 2421: 2417: 2410: 2407: 2404: 2399: 2395: 2384:), therefore 2383: 2361: 2356: 2352: 2345: 2336: 2331: 2327: 2315: 2311: 2306: 2300: 2296: 2290: 2280: 2274: 2268: 2264: 2258: 2233: 2226: 2216: 2212: 2202: 2171: 2169: 2161: 2154: 2147: 2140: 2133: 2126: 2120: 2116: 2106: 2093: 2083: 2080: 2076: 2070: 2061: 2058: 2052: 2042: 2039: 2035: 2029: 2020: 2017: 1990: 1980: 1977: 1973: 1967: 1958: 1955: 1949: 1939: 1936: 1932: 1926: 1917: 1914: 1903: 1859: 1856: 1852: 1849: 1843: 1840: 1836: 1830: 1826: 1823: 1817: 1814: 1810: 1800: 1792: 1789: 1785: 1782: 1776: 1773: 1769: 1763: 1760: 1757: 1754: 1744: 1735: 1732: 1728: 1723: 1720: 1713: 1710: 1707: 1704: 1683: 1670: 1660: 1657: 1653: 1647: 1638: 1635: 1629: 1619: 1616: 1612: 1606: 1597: 1594: 1567: 1557: 1554: 1550: 1544: 1535: 1532: 1526: 1516: 1513: 1509: 1503: 1494: 1491: 1480: 1476: 1460: 1455: 1451: 1447: 1444: 1441: 1438: 1435: 1430: 1426: 1422: 1418: 1415: 1409: 1406: 1402: 1396: 1392: 1389: 1385: 1382: 1368: 1367:quadrilateral 1352: 1349: 1346: 1341: 1337: 1333: 1329: 1326: 1322: 1319: 1307: 1301: 1300:supplementary 1296: 1278: 1274: 1271: 1265: 1262: 1258: 1252: 1248: 1245: 1239: 1236: 1232: 1221: 1205: 1202: 1199: 1195: 1192: 1186: 1183: 1179: 1173: 1170: 1167: 1164: 1137: 1126: 1123: 1119: 1109: 1106: 1099: 1088: 1085: 1081: 1071: 1068: 1060: 1056: 1053: 1049: 1044: 1041: 1034: 1031: 1028: 1025: 1011: 1007: 1000: 992: 991:in Figure 4. 986: 982: 974: 966: 959: 946: 940: 930: 926: 922: 919: 915: 912: 904: 898: 891: 885: 880: 873:Special cases 870: 864: 858: 853: 849: 844: 842: 834: 828: 821: 815: 810: 802: 796: 789: 783: 774: 769: 754: 746: 742: 738: 733: 729: 717: 713: 709: 704: 700: 693: 689: 683: 675: 671: 667: 662: 658: 646: 642: 638: 633: 629: 621: 617: 613: 607: 599: 595: 591: 586: 582: 558: 550: 546: 542: 537: 533: 521: 517: 513: 508: 504: 497: 493: 487: 479: 475: 471: 466: 462: 450: 446: 442: 437: 433: 426: 422: 416: 408: 404: 400: 395: 391: 374: 367: 357: 350: 340: 333: 326: 319: 309: 302: 297: 293: 289: 277: 270: 261: 254: 241: 234: 220: 211: 209: 205: 194: 192: 187: 185: 180: 171: 158: 154: 143: 139: 137: 133: 129: 125: 121: 117: 113: 109: 96: 92: 84: 81: 73: 63: 59: 53: 52: 46: 41: 32: 31: 19: 3355:(1): 34–46, 3352: 3348: 3332: 3312: 3305: 3281:Radical axis 3246: 3237: 3230: 3224: 3216: 3209: 3206: 3200: 3192: 3185: 3174: 3170: 3023: 3015: 3008:. Since the 2987: 2985: 2972: 2774: 2761: 2754: 2745: 2741: 2734: 2730: 2717: 2710: 2705: 2692: 2689: 2666: 2588:and radius 2581: 2575: 2569: 2304: 2298: 2294: 2288: 2278: 2272: 2266: 2262: 2256: 2238:(Figure 5). 2231: 2224: 2221: 2208: 2172: 2159: 2152: 2145: 2138: 2131: 2124: 2115:radical axis 2112: 1898:R, P, S', Q' 1684: 1305: 1294: 1009: 1005: 1001: 998: 984: 980: 972: 970: 965:radical axis 927: 923: 920: 916: 902: 896: 889: 883: 879:affine plane 876: 862: 856: 845: 840: 832: 826: 819: 813: 808: 800: 794: 787: 781: 770: 372: 365: 355: 348: 338: 331: 324: 317: 307: 300: 295: 285: 275: 268: 259: 252: 239: 232: 207: 200: 188: 176: 156: 152: 135: 131: 119: 115: 111: 105: 91: 76: 67: 48: 1477:. From the 1013:(Figure 4). 380:, this is: 62:introducing 3380:Categories 3297:References 2967:See also: 2003:Similarly 973:homologous 184:projection 45:references 3369:122408307 3318:MathWorld 3085:¯ 3061:¯ 2934:¯ 2916:⋅ 2911:¯ 2893:¯ 2875:⋅ 2870:¯ 2850:¯ 2832:¯ 2812:¯ 2794:¯ 2643:¯ 2618:¯ 2529:∠ 2502:∠ 2469:∠ 2436:∠ 2414:∠ 2392:∠ 2366:¯ 2341:¯ 2314:isosceles 2310:homothety 2254:. Extend 2089:¯ 2071:⋅ 2066:¯ 2048:¯ 2030:⋅ 2025:¯ 1986:¯ 1968:⋅ 1963:¯ 1945:¯ 1927:⋅ 1922:¯ 1860:α 1834:∠ 1808:∠ 1804:⟹ 1793:α 1767:∠ 1752:∠ 1748:⟹ 1717:△ 1714:≅ 1702:△ 1666:¯ 1648:⋅ 1643:¯ 1625:¯ 1607:⋅ 1602:¯ 1563:¯ 1545:⋅ 1540:¯ 1522:¯ 1504:⋅ 1499:¯ 1456:∘ 1445:α 1439:α 1436:− 1431:∘ 1400:∠ 1380:∠ 1350:α 1347:− 1342:∘ 1317:∠ 1256:∠ 1230:∠ 1203:α 1177:∠ 1162:∠ 1132:¯ 1114:¯ 1094:¯ 1076:¯ 1038:∠ 1023:∠ 710:− 639:− 614:− 191:chirality 3259:See also 2929:′ 2888:′ 2845:′ 2807:′ 2613:′ 2546:′ 2522:′ 2495:′ 2486:′ 2462:′ 2453:′ 2316:because 2248:P, Q, P' 2084:′ 2043:′ 1981:′ 1940:′ 1882:Segment 1853:′ 1844:′ 1827:′ 1818:′ 1786:′ 1777:′ 1736:′ 1724:′ 1661:′ 1620:′ 1558:′ 1517:′ 1419:′ 1410:′ 1393:′ 1330:′ 1291:Because 1275:′ 1266:′ 1249:′ 1240:′ 1196:′ 1187:′ 1127:′ 1089:′ 1057:′ 1045:′ 841:internal 809:external 296:opposite 136:internal 132:external 108:geometry 70:May 2024 3391:Circles 3127:(where 1365:In the 1218:By the 848:tangent 197:Circles 179:similar 157:A'B'C' 58:improve 3367: 2382:radius 2119:powers 2005:QSP'R' 1582:PRS'Q' 1370:QSR'P' 1150:since 1010:EQ'S' 47:, but 3365:S2CID 3345:(PDF) 3010:locus 2989:plane 2974:this. 2669:P, Q' 2662:Q, P' 2566:Thus 2305:P'Q' 2183:S, R' 2179:Q, P' 1295:QSR' 989:Q, P' 977:Q, Q' 954:S, R' 950:Q, P' 118:or a 3207:Let 3201:line 3152:and 3028:BB' 3020:AA', 2693:both 2250:and 2191:P'R' 1892:and 330:), ( 110:, a 3357:doi 3225:all 3133:, r 3003:, H 2999:, H 2576:P'Q 2297:, △ 2273:P' 1885:RQ' 1452:180 1427:180 1338:180 1306:ESQ 1302:to 1298:is 1008:, △ 1006:EQS 981:not 771:In 314:by 155:, ∠ 153:ABC 106:In 3382:: 3363:, 3353:22 3351:, 3347:, 3316:, 3283:, 3236:, 3215:, 3191:, 3156:AC 3149:BC 3142:AB 3026:AB 3018:AB 3005:AC 3001:BC 2997:AB 2772:. 2740:, 2716:, 2664:. 2295:PQ 2265:, 2252:Q' 2230:, 2187:QS 2151:, 2137:, 1904:, 1894:S' 1689:: 1372:, 1309:, 1222:, 895:, 825:, 793:, 371:, 354:, 337:, 323:, 306:, 274:, 258:, 238:, 3359:: 3241:2 3238:C 3234:1 3231:C 3220:2 3217:C 3213:1 3210:C 3196:2 3193:C 3189:1 3186:C 3159:. 3154:H 3147:H 3140:H 3135:B 3131:A 3129:r 3111:A 3107:r 3101:B 3097:r 3091:= 3081:A 3076:B 3073:A 3067:H 3057:B 3052:B 3049:A 3043:H 3024:H 3022:△ 3016:H 3014:△ 2995:H 2957:E 2953:E 2939:. 2926:P 2922:E 2907:Q 2904:E 2898:= 2885:Q 2881:E 2866:P 2863:E 2856:; 2842:Q 2838:E 2828:Q 2825:E 2818:= 2804:P 2800:E 2790:P 2787:E 2770:E 2765:2 2762:T 2760:/ 2758:1 2755:T 2749:2 2746:T 2744:/ 2742:C 2738:1 2735:T 2733:/ 2731:C 2726:C 2721:2 2718:T 2714:1 2711:T 2648:. 2639:Q 2634:1 2630:T 2623:= 2610:P 2604:1 2600:T 2585:1 2582:T 2573:1 2570:T 2568:△ 2554:. 2551:Q 2543:P 2537:1 2533:T 2526:= 2519:P 2515:Q 2510:1 2506:T 2499:= 2492:P 2483:Q 2477:2 2473:O 2466:= 2459:Q 2450:P 2444:2 2440:O 2433:= 2430:P 2427:Q 2422:1 2418:O 2411:= 2408:Q 2405:P 2400:1 2396:O 2380:( 2362:Q 2357:1 2353:O 2346:= 2337:P 2332:1 2328:O 2302:2 2299:O 2292:1 2289:O 2287:△ 2282:1 2279:T 2270:2 2267:O 2263:Q 2260:1 2257:O 2244:E 2240:E 2235:2 2232:O 2228:1 2225:O 2199:G 2195:G 2175:E 2163:3 2160:C 2158:/ 2156:2 2153:C 2149:3 2146:C 2144:/ 2142:1 2139:C 2135:2 2132:C 2130:/ 2128:1 2125:C 2122:( 2094:. 2081:R 2077:I 2062:S 2059:I 2053:= 2040:P 2036:I 2021:Q 2018:I 1991:. 1978:S 1974:I 1959:R 1956:I 1950:= 1937:Q 1933:I 1918:P 1915:I 1890:P 1857:= 1850:R 1841:P 1837:P 1831:= 1824:Q 1815:S 1811:R 1790:= 1783:R 1774:P 1770:I 1764:= 1761:I 1758:P 1755:R 1733:R 1729:I 1721:P 1711:R 1708:I 1705:P 1687:I 1671:. 1658:S 1654:E 1639:R 1636:E 1630:= 1617:Q 1613:E 1598:P 1595:E 1568:. 1555:R 1551:E 1536:S 1533:E 1527:= 1514:P 1510:E 1495:Q 1492:E 1461:, 1448:= 1442:+ 1423:= 1416:R 1407:P 1403:Q 1397:+ 1390:R 1386:S 1383:Q 1353:. 1334:= 1327:R 1323:S 1320:Q 1304:∠ 1293:∠ 1279:. 1272:Q 1263:S 1259:E 1253:= 1246:R 1237:P 1233:E 1206:. 1200:= 1193:Q 1184:S 1180:E 1174:= 1171:Q 1168:S 1165:E 1152:E 1138:, 1124:S 1120:E 1110:S 1107:E 1100:= 1086:Q 1082:E 1072:Q 1069:E 1061:, 1054:S 1050:E 1042:Q 1035:= 1032:S 1029:E 1026:Q 1004:△ 961:G 941:. 906:2 903:B 900:1 897:B 893:2 890:A 887:1 884:A 866:1 863:B 860:2 857:A 836:2 833:A 830:1 827:B 823:2 820:B 817:1 814:A 804:2 801:B 798:1 795:B 791:2 788:A 785:1 782:A 777:α 755:. 752:) 747:2 743:y 739:, 734:2 730:x 726:( 718:2 714:r 705:1 701:r 694:1 690:r 684:+ 681:) 676:1 672:y 668:, 663:1 659:x 655:( 647:2 643:r 634:1 630:r 622:2 618:r 608:= 605:) 600:e 596:y 592:, 587:e 583:x 579:( 559:. 556:) 551:2 547:y 543:, 538:2 534:x 530:( 522:2 518:r 514:+ 509:1 505:r 498:1 494:r 488:+ 485:) 480:1 476:y 472:, 467:1 463:x 459:( 451:2 447:r 443:+ 438:1 434:r 427:2 423:r 417:= 414:) 409:0 405:y 401:, 396:0 392:x 388:( 378:) 376:0 373:y 369:0 366:x 364:( 359:2 356:r 352:1 349:r 344:) 342:2 339:y 335:2 332:x 328:1 325:y 321:1 318:x 316:( 311:2 308:C 304:1 301:C 282:. 279:2 276:B 272:1 269:B 263:2 260:A 256:1 253:A 248:d 243:2 240:r 236:1 233:r 228:E 224:I 151:∠ 147:S 100:O 83:) 77:( 72:) 68:( 54:. 20:)

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