The altitude of one triangle is the median of the other.


Let the squares have sides a and b and consider the altitude to triangle ABC in the diagram. Then by the sine rule

a sinA = b sinB .

The altitude to triangle ABC cuts the angle at the common vertex in the other triangle into two angles equal to angles A and B and the opposite side into lengths p and q where

p = b sinA sinz = a sinB sinz = a sinB sin(p- z) = q

showing that this is the median of the triangle.