Right triangle $\triangle ABC$, $D$ lies on $AB$, inscribed in a circle of radius $9$. Find the measure of $CD$ I saw this mathematical puzzle on an Instagram post today. As title suggests, we have a right angled triangle inscribed in a circle with radius $D$ and some angles. The goal is to find the length of $CD$. I'll share my approach as an answer below. I'm not quite sure if my answer is correct, so please feel free to point out any faults in my approach and/or post your own approaches too!


Law of Tangents, let $t = \tan α$
$\displaystyle \frac{x+6}{x-6} = 
\frac{\tan \frac{(3α-90°)+(90°-α)}{2}}{\tan \frac{(3α-90°)-(90°-α)}{2}}
= \frac{\tan α}{\tan (2α-90°)} = -(\tan α)(\tan 2α) = \frac{-2t^2}{1-t^2}$
$\displaystyle \frac{x+12}{x-12} = 
\frac{\tan \frac{(180°-3α)+(α)}{2}}{\tan \frac{(180°-3α)-(α)}{2}}
= \frac{\tan (90°-α)}{\tan (90°-2α)} = \frac{\tan 2α}{\tan α} = \frac{2}{1-t^2}$
$→ \displaystyle \frac{x+6}{x-6} + \frac{x+12}{x-12} = 2$
$→ x^2-6×12 = (x-6)(x-12) $
$\displaystyle → x = \frac{6×12}{(6+12)÷2} = \frac{72}{9} = 8$
