Paralogic Triangles

At the points where a line cuts the sides of a Triangle $\Delta A_1A_2A_3$, perpendiculars to the sides are drawn, forming a Triangle $\Delta B_1B_2B_3$ similar to the given Triangle. The two triangles are also in perspective. One point of intersection of their Circumcircles is the Similitude Center, and the other is the Perspective Center. The Circumcircles meet Orthogonally.

See also Circumcircle, Orthogonal Circles, Perspective Center, Similitude Center

References

Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 258-262, 1929.