In ΔABC, ∠ACB=90° and DCEF is a square . . .

Question : In ΔABC, ∠ACB=90° and DCEF is a square, as shown in figure. Prove that DE²=AD×BF

Doubt by Afifa

Solution : 

DCEF is a square. 
DE=EF=FC=CD (All sides of square are equal)

In ΔACB and ΔADE
∠A=∠A (Common)
∠ACB=∠ADE (Each 90°)
ΔACB~ΔADE (By AA Similarity Criteria) — (1) 

In ΔACB and ΔEFB
∠B=∠B (Common)
∠ACB=∠EFB (Each 90°)
ΔACB~ΔEFB (By AA Similarity Criteria) — (2) 

From equation (1) & (2) 

ΔADE~ΔEFB 

AD/EF=ED/BF (By CPCT)
AD×BF=EF×ED 
AD×BF=EF×DE
AD×BF=DE×DE [∵EF=DE]
AD×BF=DE²
DE²=AD×BF

Hence Proved.