So, I am solving EGMO and had a confusion on this problem;
A circle has center on the side AB of the cyclic quadrilateral $ABCD$. The other three sides are tangent to the circle. Prove that $AD + BC = AB$.
My question is ;
Is it necessary that the center of circle is the midpoint of $AB$? Does it matter?
As I was able to solve most of the other problems, and from what I have heard, this is one of the easier problems so I shouldn't be having much trouble with it but I don't know why before starting the problem (at 2 in the night), I had this confusion in my mind so now I can't move past.
I am sorry if this is a dumb question, but please please clear my confusion.
Best Answer
Answer: No, it's not necessary that the center of the circle is, at the same time, the midpoint of $AB$.
For the proof, consider a point $T$ on $AB$, such that $BT=BC$, and let us employ directed angles. We will proceed to show that $AT=AD$. In fact, let $O$ be the center of the circle, and observe that $DOTC$ is a cyclic quadrilateral too: $$\angle ODC=\frac{\angle ADC}2=\frac{180^\circ-\angle CBA}2=\angle BTC$$ But then $$\angle DTA=\angle DCO=\frac{\angle DCB}2=\frac{180^\circ -\angle BAD}2\implies AT=AD$$