At the points where a line X cuts the sides of a triangle Δ A_1 A_2 A_3, draw three perpendiculars to the sides, one through each point of intersection. The resulting three lines intersect pairwise in three points that form a triangle Δ B_1 B_2 B_3 known as the paralogic triangle of Δ A_1 A_2 A_3. The paralogic and original triangles are similar triangles, and two triangles are also perspective triangles with the line X being the perspectrix. Amazingly, the circumcircles of Δ A_1 A_2 A_3 and Δ B_1 B_2 B_3 meet orthogonally in two points, with one point of intersection being their similitude center, and the other being their perspector.

circumcircle | orthogonal circles | perspector | perspectrix | similitude center | Sondat's theorem