$$ABCD$$ is a square of side length 1 unit. Two circles $$C_1$$, $$C_2$$ of equal radii are drawn inside the square such that they are touching each other and $$C_1$$ touching the sides $$AB$$, $$BC$$ and $$C_2$$ touching the sides $$CD$$, $$DA$$. Then find the radius.

Take $$A(0, 0)$$, $$B(1, 0)$$, $$C(1, 1)$$, $$D(0, 1)$$. The centres are $$(1 - r, r)$$ and $$(r, 1 - r)$$, both on diagonal $$BD$$, and their distance is $$\sqrt{2}(1 - 2r)$$. Touching circles give $$\sqrt{2}(1 - 2r) = 2r$$, so $$r = \frac{\sqrt{2}}{2 + 2\sqrt{2}} = \frac{2 - \sqrt{2}}{2} \approx 0.293$$ units.