Question 252012

So we have three triangles
Two right triangles BDA  and CDA and one to be proven is isosceles ABC.
What do we know?

BD=DC because AD bisects the triangle ABC
Angle BDA and CDA are right angles because BC is perpendicular to AD
So we have two legs of one right triangle congruent to two legs of another right triangle.

Since angle BAD is congruent to angle CAD so we have two angles of triangle BDA and CDA
Therefore the other angles must also be congruent and so the two triangles are congruent and there the third sides are congruent.
But since side AB is congruent to side AC then triangle ABC must be isosceles.