Question 52
Let $ABC$ be the isosceles triangle with $AB = AC$ and $M,N$ are the midpoints of $AB, AC$, respectively. The circle $(K)$ is tangent to line $AB$ at $B$ and tangent to $AC$ at $C$. An arbitrary circle $(I)$ tangents to $(K)$, passes through $A$ and cuts the line $MN$ at $D,E$. Prove that:
a. Point $I$ belongs to a fixed line and the angle $\angle MAN$ is a constant.
b. The circle with diameter $DE$ is always tangent to two certain fixed circles and has the center belongs to the line $AK$.
a. Point $I$ belongs to a fixed line and the angle $\angle MAN$ is a constant.
b. The circle with diameter $DE$ is always tangent to two certain fixed circles and has the center belongs to the line $AK$.