X1145

3rd EHRMANN POINT

Let Ja,Jb,Jc be the respective excenters of ABC, and let AB be the projection of A on JaJb, let AC be the projection of A on JaJc, and define BC, BA, CA, CB cyclically. The Euler lines of the three triangles JaABAC, JbBCBA, JcCACB concur in X1145. Also, X1145 is X974 of the excentral triangle. (Analogously, X442 is X973 of the excentral triangle; see the note at X442.) Jean-Pierre Ehrmann (#4200, 10/24/01)
< 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) { 8, 100 }: Central Line L(8, 100) { 3, 8 }: Central Line L(3, 8) { 10, 11 }: Central Line L(10, 11) { 9, 80 }: Central Line L(9, 80) { 144, 153 }: Central Line L(144, 153) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_8 = NAGEL POINT (0.4694, 0.1333) X_100 = ANTICOMPLEMENT OF FEUERBACH POINT (0.1429, 0.4666) X_3 = CIRCUMCENTER (0.5000, 0.1021) X_10 = SPIEKER CENTER (0.5204, 0.1833) X_11 = FEUERBACH POINT (0.7347, 0.0667) X_9 = MITTENPUNKT (0.5110, 0.1615) X_80 = REFLECTION OF INCENTER IN FEUERBACH POINT (0.8980, -0.1000) X_144 = ANTICOMPLEMENT OF X(7) (0.4317, 0.0461) X_153 = REFLECTION OF X(20) IN X(100) (-0.1020, 1.1248) X_1145 = 3rd EHRMANN POINT (0.3061, 0.2999) A B C X8 X100 X3 X10 X11 X9 X80 X144 X153 X1145 (a, b, c) = (5, 6, 7) | AB = 1

X1145 (3rd EHRMANN 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 \(f(a,b,c) = bc(b + c - 2a)[2abc - (b + c)(a^{2} - (b - c)^{2})]\)
Barycentrics: \(af(a,b,c) : bf(b,c,a) : cf(c,a,b)\)
Search Value (6, 9, 13): \(3.554662746764\)

X1145 lies on the Mandart hyperbola.

▸ Incident Central Lines8 lines

{2, 1000}, {3, 8}, {9, 80}, {10, 11}, {119, 517}, {144, 153}, {214, 519}, {484, 529}

▸ Functional Relations8 relations