Question 196
In the square $ABCD$, a circle $\mathcal{K}$, a semicircle $\mathcal{P}$ and a quarter circle $\mathcal{Q}$ are inscribed. The quarter circle $\mathcal{Q}$ has its centre at the vertex $A$ and contains the points $B$ and $D$. The semicircle $\mathcal{P}$ has its centre at the midpoint of the side $AD$ and contains the points $A$ and $D$. The circle $\mathcal{K}$ touches the semicircle $\mathcal{P}$ at the point $E$, the quarter circle $\mathcal{Q}$ at the point $F$, and the side $AB$ of the square $ABCD$ at the point $G$.
a) Express the radius of the circle $\mathcal{K}$ in terms of the side length of the square $ABCD$.
b) Prove that the line $EF$ is parallel to the side $AB$.
a) Express the radius of the circle $\mathcal{K}$ in terms of the side length of the square $ABCD$.
b) Prove that the line $EF$ is parallel to the side $AB$.