Suppose that a Simson line of a triangle is perpendicular to one of the sides of the triangle. Show that the?

Question

pole must be one of the vertices of the triangle.

Note: Simson's Theorem: Let P be any point on the circumcircle of triangle ABC and let X, Y, and Z be the feet of the perpendiculars dropped from P to line BC, AC, and AB, respectively. Then points X, Y, and Z are collinear.

Answer 1

If the presumed Simson line is perpendicular to one side and the pole is not at the opposite vertex, then the other two Simson points do not lie on the presumed Simson line and the presumption is false Thus the pole must be at the opposite vertex. I believe there is also an analytical proof showing that the Simson line approaches the perpendicular to the side as the pole approaches the opposite vertex.