The following sequence of problems (8-11) provides a proof of the validity of the Simson L...

The following sequence of problems (8-11) provides a proof of the validity of the Simson Line for a triangle. Each problem independently contains a useful, signifi-t cant geometric result.

Existence of Simson Line For any point P on the circumcircle of AABC we have, by Ptolemy's Theorem (Problem 10)

ΔB ∙ PC + BC ∙ AP = AC ∙ BP

Convert this to standard form, with x = PA, y = PB, z = PC, a = BC, etc. Use the formulas from Problem 9 to show that D, E, and F are collinear. (See figure below.)