Question 5
$a$) Acute $ABC$ has median $AM , M$ on $BC$, and altitude $BE , E$ on $AC$. $K = AM \cap BE$ and $MAC = 30 ^{\circ}$. Prove $AC = \sqrt{3} BK$. $b$) Acute $ABC$ has $C = 45 ^{\circ}$ and $AB \lt AC$, altitude $AH , H$ on $BC$, and bisector $BD , D$ on $AC$. The perpendicular bisector of $BC$ meets $AC$ at $K$. $I = BD \cap AH$ and $KA = KD$. Prove $AIB$ is isosceles.