Double Angle Formulas/Sine/Proof 3

Proof


Consider an isosceles triangle $\triangle ABC$ with base $BC$ and apex $\angle BAC = 2 \alpha$.

Construct the angle bisector to $\angle BAC$ and name it $AH$:
 * $\angle BAH = \angle CAH = \alpha$

From Bisector of Apex of Isosceles Triangle is Perpendicular to Base:
 * $AH \perp BC$

From Area of Triangle in Terms of Two Sides and Angle:

By definition of sine:

and so: