X501

MIQUEL ASSOCIATE OF INCENTER

Miquel’s theorem states that if A', B', C' are points (other than A, B, C) on sidelines BC, CA, AB, respectively, then the circles AB'C', BC'A', CA'B' meet at a point. Suppose P is a point and A' = P∩BC, B' = P∩CA, C' = P∩AB; the point in which the three circles is the Miquel associate of P. (Paul Yiu, 7/6/99)
< A B C >: Reference Triangle △ABC [ A B ]: Sideline AB (Base c = 1) [ B C ]: Sideline BC (Side a) [ C A ]: Sideline CA (Side b) { 36, 58 }: Central Line L(36, 58) { 35, 110 }: Central Line L(35, 110) { 1, 229 }: Central Line L(1, 229) { 21, 214 }: Central Line L(21, 214) { 10, 662 }: Central Line L(10, 662) { 284, 942 }: Central Line L(284, 942) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_502 = ISOGONAL CONJUGATE OF X(501) (0.5927, 0.3141) X_36 = INVERSE-IN-CIRCUMCIRCLE OF INCENTER (1.3333, 1.6330) X_58 = ISOGONAL CONJUGATE OF X(10) (0.6107, 0.2850) X_35 = {X(1),X(3)}-HARMONIC CONJUGATE OF X(36) (0.5373, 0.1706) X_110 = FOCUS OF KIEPERT PARABOLA (0.2516, 0.5478) X_1 = INCENTER (0.5714, 0.2333) X_229 = -CEVA CONJUGATE OF X(81) (0.5961, 0.2991) X_21 = SCHIFFLER POINT (0.5287, 0.1771) X_214 = -CEVA CONJUGATE OF X(44) (0.3571, 0.3499) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_662 = TRILINEAR POLE OF LINE X(1)X(21) (0.1811, 0.4951) X_284 = -CROSS CONJUGATE OF X(21) (0.5646, 0.2088) X_942 = INCIRCLE-INVERSE OF X(36) (0.5878, 0.2633) X_501 = MIQUEL ASSOCIATE OF INCENTER (0.5444, 0.1612) A B C X502 X36 X58 X35 X110 X1 X229 X21 X214 X10 X662 X284 X942 X501 (a, b, c) = (5, 6, 7) | AB = 1

X501 (MIQUEL ASSOCIATE OF INCENTER) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(f(a,b,c) : f(b,c,a) : f(c,a,b)\), where \(f(a,b,c) = a[a^{3} - b^{3} - c^{3} - bc(a + b + c) + ab(a - b) + ac(a - c)]/(b + c)\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(4.266239413525\)
▸ Incident Central Lines10 lines

{1, 229}, {10, 662}, {21, 214}, {35, 110}, {36, 58}, {215, 1364}, {284, 942}, {572, 992}, {595, 1326}, {759, 1385}

▸ Functional Relations1 relation