Finding angle associated with point inside an equilateral triangle.

Question

$\triangle{ABC}$ is an equilateral triangle.
$|AD|=6$.
$|BD|=10$.
$|CD|=8$.
What is $m\angle{CDA}$?

First thing comes to mind is Ceva theorem. I used its trigonometric form to reach somewhere. Also, i tried to reach the result with trigonometry by drawing perpendicular to sides from $D$, hoping to exploit Viviani's theorem somehow. These trigonometric attempts came, because i can't find a geometric way to prove that angle is $150^\circ$.

I prefer a geometric solution but all approaches are welcome, especially if it's more direct.

Answer 1

Rotate $\triangle ADC$ w.r.t. $A$ so that $AC$ overlaps $AB$. Suppose image of $D$ is $E$. Now $AE=AD=ED=6$ for $\angle EAD=\pi/3 $. Now in $\triangle BED, BE=8, ED=6 ,BD=10$ so $ \triangle BED$ is right angled with $\angle E=\pi/2$. Hence evaluate $\angle AEB = 5\pi/6=\angle ADC$.

PS: Please draw a picture accordingly yourself.