X1123

PAASCHE POINT

Let D and E be the congruent circles each tangent to the other and to line BC, with D also tangent to line AB and E also tangent to line CA, meeting in a point A' lying outside triangle ABC. Define B' and C' cyclically. Then A'B'C' is perspective to ABC, and the perspector is X1123. See
< 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) { 37, 158 }: Central Line L(37, 158) { 57, 482 }: Central Line L(57, 482) { 2, 586 }: Central Line L(2, 586) { 81, 1335 }: Central Line L(81, 1335) A = Vertex A (0.0000, 0.0000) B = Vertex B (1.0000, 0.0000) C = Vertex C (0.6122, 0.5999) X_1124 = ISOGONAL CONJUGATE OF X(1123) (0.5846, 0.2489) X_37 = CROSSPOINT OF INCENTER AND CENTROID (0.5567, 0.2158) X_158 = -CROSS CONJUGATE OF X(92) (0.6215, 0.5117) X_57 = ISOGONAL CONJUGATE OF X(9) (0.6138, 0.3110) X_482 = 2nd EPPSTEIN POINT (0.5835, 0.2611) X_2 = CENTROID (0.5374, 0.2000) X_586 = 2nd CONGRUENT SHRUNK INSQUARES POINT (-0.2506, -0.5117) X_81 = CEVAPOINT OF INCENTER AND SYMMEDIAN POINT (0.5849, 0.2510) X_1335 = {X(1),X(6)}-HARMONIC CONJUGATE OF X(1124) (0.4024, 0.0326) X_1123 = PAASCHE POINT (0.5572, 0.2178) A B C X1124 X37 X158 X57 X482 X2 X586 X81 X1335 X1123 (a, b, c) = (5, 6, 7) | AB = 1

X1123 (PAASCHE POINT) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

Trilinears: \(1/(1 +\sin A) : 1/(1 + \sin B) : 1/(1 + \sin C)\)
Barycentrics: \((\sin A)/(1 + \sin A) : (\sin B)/(1 + \sin B) : (\sin C)/(1 + \sin C)\)
Search Value (6, 9, 13): \(2.005660924796\)
▸ Incident Central Lines8 lines

{1, 3069}, {2, 586}, {37, 158}, {57, 482}, {81, 1335}, {498, 3302}, {499, 3300}, {920, 3068}

▸ Functional Relations3 relations

Notes

Ivan Paasche, Aufgabe P 933, Praxis der Mathematik 1 (1990), page 40.

Let PA be the parabola with focus A and directrix BC, and let LA be the line of the points of intersection of PA with the segments AB and AC. Define LB and LC cyclically. Let A' = LB∩LC, and define B' and C' cyclically. The lines AA', BB', CC' concur in X1123. (Randy Hutson, 9/23/2011)

Suppose that ABC is an acute triangle. Let \(o_{A}\) be the circle with diameter BC. Let \(w_{A}\) be the circle tangent to segments AB and AC and also externally tangent to \(o_{A}\), in point XA. Define XB and XC cyclically. The lines AXA, BXB, CXC concur in X1123. (Tomasz Cieśla, 19 January 2013) For a related construction, see X1336.

In Hutson’s construction of X1123 given above, the parabola PA meets segments AB and AC in two points, here denoted by Ab and Ac. Cyclically, the parabolas PB and PC determine four more points; the six points are then Ab =\(0 : 2R : c\), Ac =\(0 : b : 2R\), Bc =\(a : 0 : 2R\), Ba =\(2R : 0 : c\), Ca =\(2R : b : 0\), Cb =\(a : 2R : 0\). These points lie on a conic, named the Paasche conic at X37861. (Vijay Krishna, April 14, 2020)

For a construction and relationships to other points, see the preamble just before X37994.