OrthicAndPedal2

Orthic and pedal relation ÉÉ

Here are some additional remarks following the discussion in OrthicAndPedal.html . For each point X on the plane of ABC, its pedal triangle (X₁X₂X₃ not shown) and the pedal triangle (Y₁Y₂Y₃ not shown) of the isogonal conjugate Y share the same circumcircle (see Pedal.html ).
Consider now a point P on the circumcircle and reflect lines XP, YP on the sides of triangle ABC to obtain through their intersections triangles X_AX_BX_C and Y_AY_BY_C, both similar to the orthic of ABC.
[1] Triangles X_AX_BX_C, Y_AY_BY_C are line perspective with respect to the Steiner line S(P) of P on the circumcircle.
[2] Fixing P on the circumcircle and varying X (consequently and Y) the two triangle change but remain perspective with respect to the same line S(P).
[3] The trilinear poles (tripoles) {X_P, Y_P} of line S(P) with respect to the two triangles are collinear with the perspectivity center H' of the triangles.

Orthic and pedal relation ÉÉ

See Also