I'm asked to prove the following:
In a convex quadrilateral, the Thales circle of every side, (meaning the half circle, where the side is the diameter) is drawn. Prove that the four half-circles cover the entire quadrilateral.
I am given the hint that I should use the Pigeonhole Principle to prove this. However I have never applied the Pigeonhole Principle to Geometry and don't know where to start. Any help would be appreciated!
Suppose the convex quadrilateral $ABCD$. Let's $O$ any point inside $ABCD$.
We have $\widehat{AOB}+\widehat{BOC}+\widehat{COD}+\widehat{DOA} = 360°$ then there exists an angle (WOLG, suppose $\widehat{AOB}$) less than or equal to $90°$.
Because $\widehat{AOB}\le 90$ then $O$ must inside the circle where its diameter is the side $AB$.