Point K is marked in an isosceles triangle ABC with base AC on the bisector BD. Prove that triangle AKC is isosceles.

Given:
isosceles triangle ABC,
АС – base,
BD – bisector,
point K belongs to the segment BD.
Prove that triangle AKC is isosceles.
Evidence:
1. Consider an isosceles triangle ABC. The bisector of the BD is both the median and the height, then the ABP = DC.
2. Consider the right-angled triangles AKD and DKC. They have a common CD leg, BP leg = DS leg. Therefore, right-angled triangle AKD = right-angled triangle KDS along two legs. Then AK = KC and the KAD angle = KCD angle, then the AKC triangle is isosceles. Q.E.D.