August 2018 Geometry#6

$\overline{AC}^2+\overline{DC}^2-2\overline{AC}\times\overline{AD}\cos{\angle ACD}=\overline{AD}^2$

$\overline{AC}^2+\overline{AB}^2-2\overline{AC}\times\overline{AB}\cos{\angle BAC}=\overline{BC}^2$

Let $\overline{AC}=x,\angle{ACD}=\angle{BAC}=\theta$ ,then

$x^2+1600-80x\cos\theta=256...(1)$

$x^2+\space\space400-40x\cos\theta=576...(2)$

$x^2-800=896...2(2)-(1)$

$x^2=1696$