A Theorem about Simultaneous Orthological and Homological Triangles

In this paper we prove that if $P_1, P_2$ are isogonal points in the triangle $ABC$, and if $A_1B_1C_1$ and $A_2B_2C_2$ are their corresponding pedal triangles such that the triangles $ABC$ and $A_1B_1C_1$ are homological (the lines $AA_1, BB_1, CC_1$ are concurrent), then the triangles $ABC$ and $A_2B_2C_2$ are also homological.

💡 Research Summary

The paper investigates a classical configuration in Euclidean triangle geometry involving two isogonal points and their pedal triangles, and establishes a new theorem linking orthology and homology. Let (ABC) be a reference triangle and let (P_{1}) and (P_{2}) be a pair of isogonal points with respect to (ABC); that is, the lines (AP_{1}) and (AP_{2}) are symmetric about the internal angle bisector at (A), and similarly for the vertices (B) and (C). For each point we construct the pedal triangle: (A_{1}B_{1}C_{1}) is formed by dropping perpendiculars from (P_{1}) to the sides (BC, CA, AB), while (A_{2}B_{2}C_{2}) is defined analogously for (P_{2}).

The central hypothesis is that the reference triangle (ABC) and the first pedal triangle (A_{1}B_{1}C_{1}) are homological, meaning that the three lines joining corresponding vertices, (AA_{1}, BB_{1}, CC_{1}), concur at a single point (O). In classical terms, this concurrency is equivalent to the trigonometric form of Ceva’s theorem applied to the angles formed by (AP_{1}) with the sides of (ABC): \