Triangle ABC has AB = AC, and a point D on AC satisfies AD = BD = BC. Find ∠BAC.

Let ∠$A=x$. AD = BD gives ∠ABD = $x$, so the exterior angle ∠BDC = $2x$. BD = BC gives ∠$C$ = ∠BDC = $2x$, and AB = $AC$ gives ∠ABC = $2x$. Then $x+2x+2x=180$, so $x=36$°.