Let ABC be a triangle, with M the midpoint of BC, and DEF a triangle, with N the midpoint of EF. Suppose that angle BAM equals angle EDN and angle CAM equals angle FDN. Show that triangles ABC and DEF are similar.
A different way to prove this is showing that if AM=DN, then both triangles are identical.
To show this, it suffices to prove that given A, M, and the angles BAM and CAM, B and C are uniquely defined.
Consider a central symmetry, with center M. B and C are symmetric to
each other. The symmetric line to AB intersects AC at C, and this is
unique, so B is unique too.
A different way
