Square $ABCD$ has a side length of $1$. $BGD$ and $AFC$ are quarter circles, and there is an inscribed circle between the quarter circles. Two points of tangency $F$ and $G$ are labeled.
How can I prove $O$ is the center of the inscribed circle?
I was thinking to show $EFGH$ is a rectangle (or equivalently an inscribed parallelogram). It is easy to see $FG$ and $EH$ are parallel to $AD$ by symmetry, but I cannot justify why $EF$ and $HG$ are parallel to $AB$ (well I thought it was obvious, but upon further thinking it didn't seem so obvious!).
The problem of solving for the radius of the circle has two solutions on Quora. However, I was having issues justifying the steps of the proofs. For example, the first proof seems to assume $O$ is the center of the circle. And the second states the three points of tangency in the circle form an equilateral triangle (which doesn't seem obvious to me).
It is definitely true $O$ is the center, but I'm at a mental block for proving this obvious thing. Any suggestions would be much appreciated, thanks!
Disclosure: I run the YouTube channel MindYourDecisions. I will give credit by linking to this thread in the "sources" for the video/blog post.
Best Answer
$\overline{AG}$ is perpendicular to $\bigcirc O$ (at $G$) and hence $\overline{EG}$ is one of its diameters. Likewise for $\overline{DF}$. Two distinct diameters always meet at the center. Q.E.D.
Note: FareedAbiFarraj's demonstration that $\overline{OF} = \overline{OG}$ does not prove that $O$ is the center: