X3597

2nd ODEHNAL POINT

Let MA, MB, MC be the centers of the circles A″, B″, C″ used to construct X3596, respectively. Then the lines A-to-MA, B-to-MB, C-to-MC concur in X3597. For more, see the reference and sketch at X3596.
< 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) { 2, 970 }: Central Line L(2, 970) { 226, 986 }: Central Line L(226, 986) { 4, 2092 }: Central Line L(4, 2092) { 429, 2052 }: Central Line L(429, 2052) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_2 = CENTROID (0.5374, 0.2000) X_970 = CENTER OF THE APOLLONIUS CIRCLE (0.2643, -0.2872) X_226 = -CEVA CONJUGATE OF X(65) (0.5778, 0.2588) X_986 = INTERSECTION OF LINES X(1)X(3) AND X(8)X(38) (0.5588, 0.2101) X_4 = ORTHOCENTER (0.6122, 0.3958) X_2092 = DANNEELS-APOLLONIUS PERSPECTOR (0.5833, 0.2396) X_429 = EULER X(58)-5th-SUBSTITUTION POINT (0.6024, 0.3700) X_2052 = ISOGONAL CONJUGATE OF X(577) (0.6161, 0.4941) X_3597 = 2nd ODEHNAL POINT (0.5950, 0.3026) A B C X2 X970 X226 X986 X4 X2092 X429 X2052 X3597 (a, b, c) = (5, 6, 7) | AB = 1

X3597 (2nd ODEHNAL POINT) 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 \(f(a,b,c) = bc/[a^{5} + a^{4}(b + c) - a^{3}(b - c)^{2} - a^{2}(b + c)(b^{2} + c^{2}) - 2abc(b^{2} + bc + c^{2}) - 2b^{2}c^{2}(b + c)]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(-1.402788751816\)
▸ Incident Central Lines4 lines

{2, 970}, {4, 2092}, {226, 986}, {429, 2052}

Notes

Let P and Q be the intersections of line BC and the excircles radical circle. Let X = X10. Let A' be the circumcenter of triangle PQX, and define B' and C' cyclically. The lines AA', BB', CC' concur in X3597. (cf. X592, where the circle is the 1st Lemoine circle and X = X182) (Randy Hutson, January 15, 2019)