Question

Let ABCD be a unit square. Draw a quadrant of a circle with A as centre and B, D as end points of the arc. Similarly, draw a quadrant of a circle with B as centre and A, C as end points of the arc. Inscribe a circle $\Gamma$ touching the arc AC externally, the arc BD internally and also touching the side AD. Find the radius of the circle $\Gamma$.

Answer 1

Let O be the centre of $\Gamma$ and r its radius.
Draw $OP\perp AD$ and $OQ\perp AB$.
Then $OP=r$, $O{Q}^2=O{A}^2-r^2=(1-r{)}^2-r^2=1-2r$.
We also have $OB=1+r$ and $BQ=1-r$.
Using Pythagoras theorem, we get
$(1+r{)}^2=(1-r{)}^2+1-2r$.
By simplification $r=1/6$.