Let X be a point in the interior of triangle PQR.
Let a line through X intersect rays QP and QR in points A and B respectively.
Let Y be the point on line segment AB such that |BY| = |AX|.

Prove that |AB| is a minimum if and only if AB is perpendicular to QY.

(In reply to re: minimum? by JayDeeKay)

The length of AB is an unknown, and varies only inasmuch that an infinite choice of various length line segments can be drawn through a point X in the interior of the triangle that intersect QP and QR.  There is also a point Y at a distance from B equal to the distance of X and A, yet the minimum length of a line segment drawn is not dependant upon QY being perpendicular to AB. Y is also a variable and has no bearing on the length of the segment except that it is defined to exist on the segment equidistant from B as A is to X.

Posted by Dej Mar on 2007-05-10 11:27:35