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 a centre and A,C as end points of the arc. Inscribe a circle Γ touching the arc AC internally, the arc BD internally and also touching the side AB .Find the radius of circle Γ. .

Answer 1

Let O be the centre of Γ. By symmetry of O is on the perpendicular bisector of AB. Draw OE AB. Then BE = AB/2 = 1/2. If r is the radius of Γ. we see that OB = 1 - r and OE = r. Using Pythagoras’ theorem

(1 - r)² = r² + (1/2)²

Simplification gives r = 3/8.