Construction and Theorem
Let be a point in the plane, and let be a triangle. We will denote by the lines . Let denote the lines , , , respectively. Let , , be the reflections of , , over the angle bisectors of angles , , , respectively. Then lines , , concur at a point , called the isogonal conjugate of with respect to triangle .
By our constructions of the lines , , and this statement remains true after permuting . Therefore by the trigonometric form of Ceva's Theorem so again by the trigonometric form of Ceva, the lines concur, as was to be proven.
Given a nonisosceles, nonright triangle let denote the center of its circumscribed circle, and let and be the midpoints of sides and respectively. Point is located on the ray so that is similar to . Points and on rays and respectively, are defined similarly. Prove that lines and are concurrent, i.e. these three lines intersect at a point. (Source)
Let be a given point inside quadrilateral . Points and are located within such that , , , . Prove that if and only if . (Source)