in the diagram below, line BC connects points B and C on the congruent sides of isosceles triangle ADE, such that triangle ABC is isosceles with vertex angle A. If AB=10, BD=5, and DE=12, what is the length of line BC?

Method 1:
The altitude in a right triangle drawn from the vertex of the right angle to the hypotenuse is the geometric mean between the 2 segment of the hypotenuse:
32 = √([44)(CD]  → 32² = 44.CD & CD = 32²/44 = 23.27

Method 2
Triangles BCD & ACB are similar, then we can write:

BC/AC = CD/CB
32/44 = CD/32  → 32²/44 = CD & CD = 23.27