# A confusion in an IMO geometry problem.

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.

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 $$\angle DTA=\angle DCO=\frac{\angle DCB}2=\frac{180^\circ -\angle BAD}2\implies AT=AD$$

Notice: We did not require $$O$$ to be the midpoint of $$AB$$. And playing around (say: with geogebra or desmos) will show that such a configuration is possible.

• thanks so much! Feb 9, 2021 at 14:18

Attempting a sketch with Geogebra, the answer seems to be no, $$O$$ does not have to be the centre of $$AB$$.

• thank you so much, may i please know how you managed to draw this? Feb 9, 2021 at 14:18
• @Aditya_math geogebra.org/classic - draw a circle round O, choose a point A and an extended diameter through O, draw tangents from A, choose a point D on one, draw tangents from D, choose a point C on one, draw tangents from C, call where that intersects the extended diameter B, draw a circle through A, C and D, adjust the points so B falls on that second circle, clean up the picture Feb 9, 2021 at 14:49
• oh wow, thanks so much! i was not able to adjust B properly before Feb 9, 2021 at 16:01