Through a point K in the plane of a triangle Δ A B C, draw parallelians through a point as illustrated above. Then there exist four points K for which P_(A C) P_(C A) = P_(A B) P_(B A) = P_(B C) P_(C B), i.e., for which the segments of the parallels have equal length. To restrict these four points, let the length of P_(B C) P_(C B) be considered negative if P_(B C) lies on the extension of A B beyond A and P_(C B) lies on the extension of C A beyond A, and positive otherwise. Define the lengths of the other two parallelians to be signed in the analogous manner. With this sign convention, there is a unique point K for which the signed parallelians have equal length. This point is called the equal parallelians point of Δ A B C.

congruent isoscelizers point | parallelian