Proving Quadrilateral Diagonal as Circle Diameter

[mr.physics]

Here is a proof my math teacher gave us a while ago. He offered an enormous number of extra credit points if someone could solve it but no one did. We later asked him for the solution and he admitted that he could not solve it either. And here it is:

A quadrilateral whose consecutive sides have lengths 39, 52, 60 and 25 respectively, is inscribed in a circle. Prove that one of the diagonals of the quadrilateral is a diameter of the circle.

 

[Elucidus]

This is a consequence of Thales' Theorem.


[Full solution snipped]

See the accompanying illustration for motivation.


--Elucidus

EDIT: I just realized this is in Homework Questions. My apologies. I have redacted the solution.