X1487

NAPOLEON CEVAPOINT

Let N denote the nine-point center, X5. Let NA = N-of-triangle NBC, and define NB and NC cyclically. Triangle NANBNC is perspective to ABC, and X1487 is the perspector. X1487 is the cevapoint of the Napoleon points, X17 and X18. (Coordinates found by Paul Yiu.)
< 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) { 4, 252 }: Central Line L(4, 252) { 140, 930 }: Central Line L(140, 930) { 5, 1173 }: Central Line L(5, 1173) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_17 = 1st NAPOLEON POINT (0.5666, 0.2383) X_18 = 2nd NAPOLEON POINT (0.4434, 0.1317) X_4 = ORTHOCENTER (0.6122, 0.3958) X_252 = ISOGONAL CONJUGATE OF X(143) (0.4565, 0.1253) X_140 = MIDPOINT OF X(3) AND X(5) (0.5281, 0.1755) X_930 = ANTICOMPLEMENT OF X(137) (0.0497, 0.3422) X_5 = NINE-POINT CENTER (0.5561, 0.2489) X_1173 = 1st SARAGOSSA POINT OF X(54) (0.6043, 0.2952) X_1487 = NAPOLEON CEVAPOINT (0.4925, 0.1879) A B C X17 X18 X4 X252 X140 X930 X5 X1173 X1487 (a, b, c) = (5, 6, 7) | AB = 1

X1487 (NAPOLEON CEVAPOINT) 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 \(f(A,B,C) = (\csc A)/[(3 - \cot B \cot C)(3 - \cot^{2} A)]\)
Barycentrics: \((\sin A)f(A,B,C) : (\sin B)f(B,C,A) : (\sin C)f(C,A,B)\)
Search Value (6, 9, 13): \(-0.103455390878\)
▸ Incident Central Lines3 lines

{4, 252}, {5, 1173}, {140, 930}

▸ Functional Relations4 relations

Notes

The construction just given for X1487 shows that it is a solution X of the “four-triangle problem” posed in

C. Kimberling, “Triangle centers as functions,” Rocky Mountain Journal of Mathematics 23 (1993) 1269-1286. See Section 5; a complete solution to the problem remains to be found.