ABC is a triangle in which ∠BAC = 90° and DEFG is a square, prove that DE² = BD × EC.

Given: ABC is a triangle in which ∠BAC = 90° and DEFG is a square.
To prove: DE² = BD × EC.
Proof: In Δ AGF and Δ DBG,
∠AGF = ∠GBD (corresponding angles)
∠GAF = ∠BDG (each = 90° )
∴ ΔAGF ∼ ΔDBG. ----------------------(i)
Similarly, ΔAFG ∼ ΔECF (AA Similarity)----------------(ii)
From (i) and (ii), ΔDBG ∼ ΔECF.


EF × DG = BD × EC. ---------------------(iii)
Also DEFG is a square ⇒ DE = EF = FG = DG ---------------(iv)
From (iii) and (iv),   DE² = BD × EC.