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 internally, the arc BD internally and also touching the side AB. Find the radius of the circle $\Gamma$.

Answer 1

Let O be the centre of $\Gamma$.
By symmetry O is on the perpendicular bisector of AB. Draw $OE\perp AB$.
Then $BE=AB/2=1/2$.
If r is the radius of $\Gamma$, we see that OB = 1 - r, and OE = r. Using Pythagoras' theorem
${(1-r)}^2=r^2+{\left(\frac{1}{2}\right)}^2$
Simplification gives $r=\frac{3}{8}$.