Question

In triangle ABC, AB = AC and BD is perpendicular to AC.

Prove that: BD² − CD² = 2CD × AD

Answer 1

Use the Pythagoras Theorem in triangle BDC

BD² = BC² − CD²

BC = CD + AD ⇒ BC² = (CD + AD)² = CD² + 2CD ⋅ AD + AD²

BD² = BC² − CD² = (CD² + 2CD ⋅ AD + AD²) − CD² = 2CD ⋅ AD + AD²

Now subtract CD²:

BD² − CD² = 2CD ⋅ AD