Proving Quadrilateral Diagonal as Circle Diameter

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A quadrilateral with sides measuring 39, 52, 60, and 25 is inscribed in a circle, leading to the challenge of proving that one of its diagonals is a diameter. This proof is linked to Thales' Theorem, which states that if a triangle is inscribed in a circle, the angle subtended by the diameter is a right angle. Despite the promise of extra credit, neither the teacher nor the students were able to solve the problem initially. An illustration was provided to aid understanding, but the full solution was later redacted from the discussion. The topic highlights the intersection of geometry and theorems in proving properties of cyclic quadrilaterals.
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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.
 
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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.
 

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