623:
558:
20:
205:
1228:. The remaining six intersection points of these nine-point circles each concur with the midpoints of the four triangles. Remarkably, there exists a unique nine-point conic, centered at the centroid of these four arbitrary points, that passes through all seven points of intersection of these nine-point circles. Furthermore, because of the Feuerbach conic theorem mentioned above, there exists a unique rectangular
353:
293:
28:
1193:
The centers of the incircle and excircles of a triangle form an orthocentric system. The nine-point circle created for that orthocentric system is the circumcircle of the original triangle. The feet of the altitudes in the orthocentric system are the vertices of the original
1270:. Consequently, the four nine-point centers are cyclic and lie on a circle congruent to the four nine-point circles that is centered at the anticenter of the cyclic quadrilateral. Furthermore, the cyclic quadrilateral formed from the four nine-pont centers is
1565:
1185:
942:
792:
1965:
and three "diagonal points" where opposite sides of the quadrilateral intersect. There are six "sidelines" in the quadrilateral; the nine-point conic intersects the midpoints of these and also includes the diagonal points. The conic is an
1913:
1729:
257:
did not entirely discover the nine-point circle, but rather the six-point circle, recognizing the significance of the midpoints of the three sides of the triangle and the feet of the altitudes of that triangle.
1022:
1366:
537:
453:
2382:
by Clark
Kimberling. The nine-point center is indexed as X(5), the Feuerbach point, as X(11), the center of the Kiepert hyperbola as X(115), and the center of the Jeřábek hyperbola as X(125).
2278:
1266:
of the cyclic quadrilateral. The nine-point circles are all congruent with a radius of half that of the cyclic quadrilateral's circumcircle. The nine-point circles form a set of four
1232:, centered at the common intersection point of the four nine-point circles, that passes through the four original arbitrary points as well as the orthocenters of the four triangles.
1412:
1057:
639:
is given, then the four triangles formed by any combination of three distinct points of that system all share the same nine-point circle. This is a consequence of symmetry: the
368:
817:
667:
277:
himself proved the existence of the circle. He was the first to recognize the added significance of the three midpoints between the triangle's vertices and the orthocenter. (
320:... the circle which passes through the feet of the altitudes of a triangle is tangent to all four circles which in turn are tangent to the three sides of the triangle...
2284:
Eigenschaften einiger merkwürdigen Punkte des geradlinigen
Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung
647:
from that second triangle. A third midpoint lies on their common side. (The same 'midpoints' defining separate nine-point circles, those circles must be concurrent.)
2386:
1760:
1582:
610:
that pass through the vertices of a triangle lies on its nine-point circle. Examples include the well-known rectangular hyperbolas of
2491:
611:
362:
A nine-point circle bisects a line segment going from the corresponding triangle's orthocenter to any point on its circumcircle.
1935:
and the nine-point circle is an instance of the general nine-point conic that has been constructed with relation to a triangle
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1315:
1263:
486:
405:
2352:
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2453:
1993:
is in one of the three adjacent regions, and the hyperbola is rectangular when P lies on the circumcircle of
75:
Note that the construction still works even if the orthocenter and circumcenter fall outside of the triangle.
1560:{\displaystyle {\frac {(b^{2}-c^{2})^{2}}{a}}:{\frac {(c^{2}-a^{2})^{2}}{b}}:{\frac {(a^{2}-b^{2})^{2}}{c}}}
1180:{\displaystyle {\overline {PA}}^{2}+{\overline {PB}}^{2}+{\overline {PC}}^{2}+{\overline {PH}}^{2}=4R^{2}.}
245:
two of the altitudes have feet outside the triangle, but these feet still belong to the nine-point circle.
937:{\displaystyle {\overline {PA}}^{2}+{\overline {PB}}^{2}+{\overline {PC}}^{2}+{\overline {PH}}^{2}=K^{2},}
787:{\displaystyle {\overline {NA}}^{2}+{\overline {NB}}^{2}+{\overline {NC}}^{2}+{\overline {NH}}^{2}=3R^{2}}
553:. The point of intersection of the bimedians of the cyclic quadrilateral belongs to the nine-point circle.
2466:
Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.
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2298:
2108:
1271:
2315:
2151:
2112:
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The nine-point circle of a reference triangle is the circumcircle of both the reference triangle's
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Discusses the nine-point circle with regard to three different quadratic forms (blue, red, green).
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8:
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2018:
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557:
241:, six of the points (the midpoints and altitude feet) lie on the triangle itself; for an
119:
42:
2447:
2345:"New applications of method of complex numbers in the geometry of cyclic quadrilaterals"
2202:"New applications of method of complex numbers in the geometry of cyclic quadrilaterals"
136:(where the three altitudes meet; these line segments lie on their respective altitudes).
2332:
2256:
College
Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle
2182:
2028:
1908:{\displaystyle x^{2}\sin 2A+y^{2}\sin 2B+z^{2}\sin 2C-2(yz\sin A+zx\sin B+xy\sin C)=0.}
2396:
1751:
be a variable point in trilinear coordinates, an equation for the nine-point circle is
300:
In 1822 Karl
Feuerbach discovered that any triangle's nine-point circle is externally
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2186:
2013:
1400:
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266:
223:
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The diagram above shows the nine significant points of the nine-point circle. Points
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70:
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of one triangle adjacent to a vertex that is an orthocenter to another triangle are
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2174:
2085:
1926:
100:
96:
1201:
are given that do not form an orthocentric system, then the nine-point circles of
2371:
1982:
1724:{\displaystyle \cos(A)\sin ^{2}(B-C):\cos(B)\sin ^{2}(C-A):\cos(C)\sin ^{2}(A-B)}
1267:
600:
596:
331:
327:
301:
273:
had stated and proven the same theorem.) But soon after
Feuerbach, mathematician
242:
2165:
Fraivert, David (July 2019). "New points that belong to the nine-point circle".
650:
Consequently, these four triangles have circumcircles with identical radii. Let
599:(with vertices at the midpoints of the sides of the reference triangle) and its
2415:
2023:
1390:
1221:
238:
153:
2477:
Ein einfacher Beweis für den Satz von
Feuerbach mit koordinatenfreien Vektoren
2385:
History about the nine-point circle based on J.S. MacKay's article from 1892:
2485:
2089:
1932:
115:
2442:
2433:
2313:
Fraivert, David (2019), "New points that belong to the nine-point circle",
805:
626:
The nine point circle and the 16 tangent circles of the orthocentric system
274:
161:
125:
56:
19:
2009:
1229:
1044:
the nine-point center. Furthermore the nine-point circle is the locus of
133:
46:
2328:
2178:
1378:
of lines passing through the circumcenter lie on the nine-point circle.
466:
55: Line segments perpendicular to the side midpoints (concur at the
2469:
2420:
2401:
1375:
330:
at which the incircle and the nine-point circle touch is called the
204:
658:
be an arbitrary point in the plane of the orthocentric system. Then
309:
305:
284:.) Thus, Terquem was the first to use the name nine-point circle.
108:
92:
80:
603:(with vertices at the feet of the reference triangle's altitudes).
1967:
1573:
Trilinear coordinates for the center of the Jeřábek hyperbola are
618:
and
Feuerbach. This fact is known as the Feuerbach conic theorem.
2306:
2267:
381:
of the nine-point circle bisects a segment from the orthocenter
2124:
296:
The nine-point circle is tangent to the incircle and excircles.
222:
are the midpoints of the line segments between each altitude's
88:
27:
2120:(4th ed.). London: Longmans, Green, & Co. p. 58.
1946:, where the particular nine-point circle instance arises when
367:
352:
214:
are the midpoints of the three sides of the triangle. Points
2071:
337:
292:
95:. It is so named because it passes through nine significant
2391:
347:
is twice the radius of that triangle's nine-point circle.
549:
be the nine-point circle of the diagonal triangle of a
1381:
A triangle's circumcircle, its nine-point circle, its
1017:{\displaystyle {\tfrac {1}{2}}{\sqrt {K^{2}-3R^{2}}}.}
970:
218:
are the feet of the altitudes of the triangle. Points
2372:"A Javascript demonstration of the nine point circle"
1763:
1585:
1415:
1318:
1060:
968:
820:
670:
489:
408:
2273:
2130:
1361:{\displaystyle {\overline {ON}}=2{\overline {NM}}.}
532:{\displaystyle {\overline {HN}}=3{\overline {NG}}.}
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1907:
1723:
1559:
1360:
1179:
1016:
936:
786:
531:
448:{\displaystyle {\overline {ON}}={\overline {NH}}.}
447:
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2253:
2237:
2225:
2098:) give a proof of the Nine-Point Circle Theorem.
2047:
2460:Nine-point conic and Euler line generalization
2149:Posamentier, Alfred S., and Lehmann, Ingmar.
1298:lies on the line connecting the circumcenter
2107:
654:represent the common nine-point center and
253:Although he is credited for its discovery,
198:
2117:A Sequel to the First Six Books of Euclid
338:Other properties of the nine-point circle
2342:
2312:
2287:(Monograph ed.), Nürnberg: Wiessner
2199:
2164:
621:
556:
291:
230:) and the triangle's orthocenter (point
140:The nine-point circle is also known as
26:
18:
2456:from the Wolfram Demonstrations Project
2448:Special lines and circles in a triangle
2072:Kocik, Jerzy; Solecki, Andrzej (2009).
2484:
1274:to the reference cyclic quadrilateral
99:defined from the triangle. These nine
91:that can be constructed for any given
2411:
2392:
2094:Kocik and Solecki (sharers of a 2010
2454:Interactive Nine Point Circle applet
2145:
2143:
2141:
2139:
954:is kept constant, then the locus of
583:belongs to the nine-point circle of
579:of intersection of the bimedians of
392:(making the orthocenter a center of
2292:
2059:
1957:. The vertices of the triangle and
465:is one-fourth of the way along the
13:
287:
16:Circle constructed from a triangle
14:
2503:
2379:Encyclopedia of Triangles Centers
2365:
2353:International Journal of Geometry
2279:Buzengeiger, Carl Heribert Ignatz
2254:Altshiller-Court, Nathan (1925),
2209:International Journal of Geometry
2136:
1920:
1243:, then the nine-point circles of
265:.) (At a slightly earlier date,
2387:History of the Nine Point Circle
2131:Feuerbach & Buzengeiger 1822
366:
351:
203:
2470:N J Wildberger. Chromogeometry.
2231:
1931:The circle is an instance of a
1036:for the corresponding constant
2492:Circles defined for a triangle
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2101:
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1419:
1385:, and the circumcircle of its
308:and internally tangent to its
1:
2475:Stefan Götz, Franz Hofbauer:
2247:
2115:Nine-Point Circle Theorem, in
2012:, a related construction for
2439:Feuerbach's Theorem: a Proof
1350:
1329:
1147:
1122:
1097:
1072:
907:
882:
857:
832:
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707:
682:
571:is the diagonal triangle of
521:
500:
437:
419:
248:
111:of each side of the triangle
7:
2003:
564:is a cyclic quadrilateral.
343:The radius of a triangle's
10:
2508:
2299:Holt, Rinehart and Winston
2258:(2nd ed.), New York:
2074:"Disentangling a Triangle"
1924:
1294:and its homothetic center
312:; this result is known as
2464:Dynamic Geometry Sketches
2430:Nine Point Circle in Java
2155:, Prometheus Books, 2012.
1985:with the triangle, but a
1197:If four arbitrary points
304:to that triangle's three
2343:Fraivert, David (2018),
2316:The Mathematical Gazette
2200:Fraivert, David (2018).
2167:The Mathematical Gazette
2152:The Secrets of Triangles
2090:10.4169/193009709x470065
2034:
958:is a circle centered at
2275:Feuerbach, Karl Wilhelm
1981:or in a region sharing
1220:concur at a point, the
199:Nine significant points
132:of the triangle to the
2293:Kay, David C. (1969),
2238:Altshiller-Court (1925
2226:Altshiller-Court (1925
2048:Altshiller-Court (1925
1963:complete quadrilateral
1950:is the orthocenter of
1909:
1725:
1561:
1399:for the center of the
1362:
1239:are given that form a
1181:
1018:
938:
788:
627:
608:rectangular hyperbolas
591:
533:
461:The nine-point center
449:
322:
297:
255:Karl Wilhelm Feuerbach
146:Karl Wilhelm Feuerbach
76:
24:
1910:
1726:
1562:
1397:Trilinear coordinates
1363:
1182:
1019:
939:
789:
625:
560:
534:
450:
318:
295:
226:intersection (points
30:
22:
2096:Lester R. Ford Award
1987:nine-point hyperbola
1761:
1583:
1413:
1316:
1241:cyclic quadrilateral
1058:
966:
818:
668:
551:cyclic quadrilateral
487:
406:
271:Jean-Victor Poncelet
191:. Its center is the
170:twelve-points circle
124:The midpoint of the
35: Triangle sides
2397:"Nine-Point Circle"
2329:10.1017/mag.2019.53
2179:10.1017/mag.2019.53
2078:Amer. Math. Monthly
2062:, pp. 18, 245)
2050:, pp. 103–110)
1942:and a fourth point
1387:tangential triangle
633:orthocentric system
473:to the orthocenter
314:Feuerbach's theorem
181:medioscribed circle
2413:Weisstein, Eric W.
2394:Weisstein, Eric W.
2260:Barnes & Noble
2029:Synthetic geometry
2014:circular triangles
1905:
1721:
1557:
1358:
1302:to the anticenter
1177:
1014:
979:
934:
784:
628:
606:The center of all
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529:
469:from the centroid
445:
316:. He proved that:
298:
279:See Fig. 1, points
260:See Fig. 1, points
142:Feuerbach's circle
77:
25:
1555:
1507:
1459:
1401:Kiepert hyperbola
1353:
1332:
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1075:
1040:, collapses onto
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396:to both circles):
267:Charles Brianchon
195:of the triangle.
193:nine-point center
166:six-points circle
85:nine-point circle
71:nine-point center
69:(centered at the
67:Nine-point circle
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2426:
2425:
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2361:
2349:
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2323:(557): 222–232,
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2295:College Geometry
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2270:
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2217:
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2173:(557): 222–232.
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2105:
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2019:Lester's theorem
1999:
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1927:Nine-point conic
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263:D, E, F, G, H, I
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189:circum-midcircle
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158:Terquem's circle
97:concyclic points
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40:
34:
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2496:
2482:
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2450:by Walter Fendt
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2006:
1994:
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1983:vertical angles
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1974:is interior to
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1268:Johnson circles
1244:
1236:
1235:If four points
1225:
1202:
1198:
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635:of four points
601:orthic triangle
597:medial triangle
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332:Feuerbach point
328:triangle center
290:
288:Tangent circles
281:
262:
251:
243:obtuse triangle
231:
227:
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211:
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174:
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62:
60:
52:
50:
45:(concur at the
38:
36:
32:
23:The nine points
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12:
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5:
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2366:External links
2364:
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2290:
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2243:
2242:
2240:, p. 241)
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2157:
2135:
2123:
2100:
2084:(3): 228–237.
2064:
2052:
2039:
2038:
2036:
2033:
2032:
2031:
2026:
2024:Poncelet point
2021:
2016:
2005:
2002:
1925:Main article:
1922:
1921:Generalization
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1877:
1874:
1871:
1868:
1865:
1862:
1859:
1856:
1853:
1850:
1847:
1844:
1841:
1838:
1835:
1832:
1829:
1826:
1821:
1817:
1813:
1810:
1807:
1804:
1801:
1796:
1792:
1788:
1785:
1782:
1779:
1776:
1771:
1767:
1753:
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1734:
1733:
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1700:
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1600:
1597:
1594:
1591:
1588:
1575:
1574:
1570:
1569:
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1554:
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1525:
1521:
1517:
1511:
1506:
1500:
1496:
1490:
1486:
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1477:
1473:
1469:
1463:
1458:
1452:
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1438:
1434:
1429:
1425:
1421:
1405:
1404:
1394:
1379:
1371:
1370:
1369:
1368:
1357:
1352:
1348:
1345:
1339:
1336:
1331:
1327:
1324:
1308:
1307:
1262:concur at the
1233:
1222:Poncelet point
1195:
1190:
1189:
1188:
1187:
1176:
1171:
1167:
1163:
1160:
1155:
1149:
1145:
1142:
1135:
1130:
1124:
1120:
1117:
1110:
1105:
1099:
1095:
1092:
1085:
1080:
1074:
1070:
1067:
1050:
1049:
1013:
1006:
1002:
998:
995:
990:
986:
977:
974:
962:with a radius
947:
946:
945:
944:
933:
928:
924:
920:
915:
909:
905:
902:
895:
890:
884:
880:
877:
870:
865:
859:
855:
852:
845:
840:
834:
830:
827:
810:
809:
804:is the common
797:
796:
795:
794:
781:
777:
773:
770:
765:
759:
755:
752:
745:
740:
734:
730:
727:
720:
715:
709:
705:
702:
695:
690:
684:
680:
677:
660:
659:
648:
620:
619:
604:
555:
554:
542:
541:
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539:
528:
523:
519:
516:
510:
507:
502:
498:
495:
479:
478:
458:
457:
456:
455:
444:
439:
435:
432:
426:
421:
417:
414:
398:
397:
364:
363:
349:
348:
339:
336:
289:
286:
250:
247:
239:acute triangle
200:
197:
154:Leonhard Euler
150:Euler's circle
138:
137:
122:
112:
61:
51:
37:
31:
15:
9:
6:
4:
3:
2:
2504:
2493:
2490:
2489:
2487:
2478:
2474:
2471:
2468:
2465:
2461:
2458:
2455:
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2437:
2435:
2431:
2428:
2423:
2422:
2417:
2414:
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2398:
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2359:
2355:
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2341:
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2330:
2326:
2322:
2318:
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2311:
2308:
2304:
2300:
2296:
2291:
2286:
2285:
2280:
2276:
2272:
2269:
2265:
2261:
2257:
2252:
2251:
2239:
2234:
2228:, p. 98)
2227:
2222:
2214:
2210:
2203:
2196:
2188:
2184:
2180:
2176:
2172:
2168:
2161:
2154:
2153:
2146:
2144:
2142:
2140:
2132:
2127:
2119:
2118:
2114:
2110:
2104:
2097:
2091:
2087:
2083:
2079:
2075:
2068:
2061:
2056:
2049:
2044:
2040:
2030:
2027:
2025:
2022:
2020:
2017:
2015:
2011:
2008:
2007:
2001:
1998:
1988:
1984:
1979:
1969:
1964:
1955:
1940:
1934:
1933:conic section
1928:
1902:
1899:
1893:
1890:
1887:
1884:
1881:
1878:
1875:
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1869:
1866:
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1756:
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1754:
1749:
1745:
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1456:
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1273:
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1252:
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1242:
1234:
1231:
1223:
1218:
1214:
1210:
1206:
1196:
1192:
1191:
1174:
1169:
1165:
1161:
1158:
1153:
1143:
1140:
1133:
1128:
1118:
1115:
1108:
1103:
1093:
1090:
1083:
1078:
1068:
1065:
1054:
1053:
1052:
1051:
1032:the locus of
1011:
1004:
1000:
996:
993:
988:
984:
975:
972:
949:
948:
931:
926:
922:
918:
913:
903:
900:
893:
888:
878:
875:
868:
863:
853:
850:
843:
838:
828:
825:
814:
813:
812:
811:
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799:
798:
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775:
771:
768:
763:
753:
750:
743:
738:
728:
725:
718:
713:
703:
700:
693:
688:
678:
675:
664:
663:
662:
661:
649:
646:
642:
634:
630:
629:
624:
617:
613:
609:
605:
602:
598:
594:
593:
588:
569:
559:
552:
544:
543:
526:
517:
514:
508:
505:
496:
493:
483:
482:
481:
480:
468:
460:
459:
442:
433:
430:
424:
415:
412:
402:
401:
400:
399:
395:
388:
376:
375:
374:
373:
369:
361:
360:
359:
358:
354:
346:
342:
341:
335:
333:
329:
324:
321:
317:
315:
311:
307:
303:
294:
285:
280:
276:
272:
268:
261:
256:
246:
244:
240:
235:
225:
208:
206:
196:
194:
190:
186:
182:
178:
177:-point circle
171:
167:
163:
159:
155:
151:
147:
143:
135:
131:
127:
123:
121:
117:
113:
110:
106:
105:
104:
102:
98:
94:
90:
86:
82:
72:
68:
58:
48:
44:
29:
21:
2443:cut-the-knot
2434:cut-the-knot
2419:
2400:
2378:
2374:at rykap.com
2357:
2351:
2320:
2314:
2297:, New York:
2294:
2283:
2255:
2233:
2221:
2212:
2208:
2195:
2170:
2166:
2160:
2150:
2126:
2116:
2113:
2103:
2081:
2077:
2067:
2055:
2043:
1996:
1989:occurs when
1977:
1961:determine a
1953:
1938:
1930:
1747:
1743:
1739:
1383:polar circle
1258:
1254:
1250:
1246:
1216:
1212:
1208:
1204:
806:circumradius
644:
640:
586:
575:. The point
567:
387:circumcenter
371:
365:
356:
350:
345:circumcircle
325:
323:
319:
299:
278:
275:Olry Terquem
259:
252:
236:
209:
202:
188:
184:
180:
173:
169:
165:
162:Olry Terquem
157:
149:
141:
139:
126:line segment
84:
78:
66:
57:circumcenter
2416:"Orthopole"
2109:Casey, John
2010:Hart circle
1230:circumconic
1028:approaches
377:The center
134:orthocenter
47:orthocenter
2248:References
2215:(1): 5–16.
1272:homothetic
1264:anticenter
1237:A, B, C, D
1226:A, B, C, D
1199:A, B, C, D
637:A, B, C, H
467:Euler line
185:mid circle
128:from each
2421:MathWorld
2402:MathWorld
2360:(1): 5–16
2337:213935239
2187:213935239
2060:Kay (1969
1891:
1873:
1855:
1837:−
1828:
1803:
1778:
1713:−
1704:
1682:
1667:−
1658:
1636:
1621:−
1612:
1590:
1529:−
1481:−
1433:−
1376:orthopole
1351:¯
1330:¯
1194:triangle.
1148:¯
1123:¯
1098:¯
1073:¯
1048:such that
994:−
908:¯
883:¯
858:¯
833:¯
758:¯
733:¯
708:¯
683:¯
522:¯
501:¯
438:¯
420:¯
306:excircles
249:Discovery
43:Altitudes
2486:Category
2307:69012075
2281:(1822),
2268:52013504
2111:(1886).
2004:See also
1746: :
1742: :
1737:Letting
808:; and if
645:segments
394:dilation
372:Figure 4
357:Figure 3
310:incircle
120:altitude
118:of each
109:midpoint
93:triangle
81:geometry
1968:ellipse
1292:
1280:
616:Jeřábek
612:Keipert
385:to the
302:tangent
282:J, K, L
237:For an
228:A, B, C
220:J, K, L
216:G, H, I
212:D, E, F
187:or the
164:), the
160:(after
152:(after
144:(after
2335:
2305:
2266:
2185:
1391:coaxal
950:where
800:where
631:If an
224:vertex
183:, the
179:, the
172:, the
168:, the
130:vertex
101:points
89:circle
83:, the
65:
63:
53:
41:
39:
33:
2348:(PDF)
2333:S2CID
2205:(PDF)
2183:S2CID
2035:Notes
1970:when
1306:where
641:sides
103:are:
87:is a
2303:LCCN
2264:LCCN
1389:are
1374:The
1276:ABCD
581:ABCD
573:ABCD
562:ABCD
545:Let
326:The
269:and
156:),
116:foot
114:The
107:The
2462:at
2441:at
2432:at
2325:doi
2321:103
2175:doi
2171:103
2086:doi
2082:116
1997:ABC
1978:ABC
1954:ABC
1939:ABC
1888:sin
1870:sin
1852:sin
1825:sin
1800:sin
1775:sin
1695:sin
1679:cos
1649:sin
1633:cos
1603:sin
1587:cos
1403:are
1259:DAB
1257:, △
1255:CDA
1253:, △
1251:BCD
1249:, △
1247:ABC
1224:of
1217:DAB
1215:, △
1213:CDA
1211:, △
1209:BCD
1207:, △
1205:ABC
1024:As
587:EFG
568:EFG
234:).
148:),
79:In
2488::
2418:.
2399:.
2356:,
2350:,
2331:,
2319:,
2301:,
2277:;
2262:,
2211:.
2207:.
2181:.
2169:.
2138:^
2080:.
2076:.
2000:.
1903:0.
614:,
334:.
2424:.
2405:.
2358:7
2327::
2289:.
2213:7
2189:.
2177::
2133:.
2092:.
2088::
1995:△
1991:P
1976:△
1972:P
1959:P
1952:△
1948:P
1944:P
1937:△
1900:=
1897:)
1894:C
1885:y
1882:x
1879:+
1876:B
1867:x
1864:z
1861:+
1858:A
1849:z
1846:y
1843:(
1840:2
1834:C
1831:2
1820:2
1816:z
1812:+
1809:B
1806:2
1795:2
1791:y
1787:+
1784:A
1781:2
1770:2
1766:x
1748:z
1744:y
1740:x
1719:)
1716:B
1710:A
1707:(
1699:2
1691:)
1688:C
1685:(
1676::
1673:)
1670:A
1664:C
1661:(
1653:2
1645:)
1642:B
1639:(
1630::
1627:)
1624:C
1618:B
1615:(
1607:2
1599:)
1596:A
1593:(
1553:c
1547:2
1543:)
1537:2
1533:b
1524:2
1520:a
1516:(
1510::
1505:b
1499:2
1495:)
1489:2
1485:a
1476:2
1472:c
1468:(
1462::
1457:a
1451:2
1447:)
1441:2
1437:c
1428:2
1424:b
1420:(
1393:.
1356:.
1347:M
1344:N
1338:2
1335:=
1326:N
1323:O
1304:M
1300:O
1296:N
1289:2
1286:/
1283:1
1245:△
1203:△
1175:.
1170:2
1166:R
1162:4
1159:=
1154:2
1144:H
1141:P
1134:+
1129:2
1119:C
1116:P
1109:+
1104:2
1094:B
1091:P
1084:+
1079:2
1069:A
1066:P
1046:P
1042:N
1038:K
1034:P
1030:N
1026:P
1012:.
1005:2
1001:R
997:3
989:2
985:K
976:2
973:1
960:N
956:P
952:K
932:,
927:2
923:K
919:=
914:2
904:H
901:P
894:+
889:2
879:C
876:P
869:+
864:2
854:B
851:P
844:+
839:2
829:A
826:P
802:R
780:2
776:R
772:3
769:=
764:2
754:H
751:N
744:+
739:2
729:C
726:N
719:+
714:2
704:B
701:N
694:+
689:2
679:A
676:N
656:P
652:N
590:.
585:△
577:T
566:△
547:ω
527:.
518:G
515:N
509:3
506:=
497:N
494:H
477::
475:H
471:G
463:N
443:.
434:H
431:N
425:=
416:N
413:O
390:O
383:H
379:N
258:(
232:S
175:n
73:)
59:)
49:)
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