Question 1
a) Acute triangle $ABC$ has median $AM$ ($M\in BC$) and altitude $BE$ ($E\in AC$). They meet at $K$, and $\angle MAC=30^\circ$. Prove that $AC=\sqrt3\,BK$.
b) Acute triangle $ABC$ has $\angle ACB=45^\circ$ and $AB\lt AC$. Draw altitude $AH$ ($H\in BC$) and angle bisector $BD$ ($D\in AC$). The perpendicular bisector of $BC$ meets $AC$ at $K$, and $BD$ meets $AH$ at $I$. Given $KA=KD$, prove that $AIB$ is isosceles.
b) Acute triangle $ABC$ has $\angle ACB=45^\circ$ and $AB\lt AC$. Draw altitude $AH$ ($H\in BC$) and angle bisector $BD$ ($D\in AC$). The perpendicular bisector of $BC$ meets $AC$ at $K$, and $BD$ meets $AH$ at $I$. Given $KA=KD$, prove that $AIB$ is isosceles.