Question

In the given figure, ∠ADE = ∠ACB and `(AD)/(DB) = (AE)/(EC)`. Prove that ΔABC is an isosceles triangle.

Answer 1

Given: ∠ADE = ∠ACB

`(AD)/(DB) = (AE)/(EC)`

To prove: ΔABC is an isosceles triangle.

Proof: `(AD)/(DB) = (AE)/(EC)`   ...(Given)

If a line divides any two sides of a triangle in the same ratio, then the line is parallel to the third side.

∴ DE || BC

⇒ ∠ADE = ∠ABC   ...(1) (Corresponding angles)

⇒ ∠AED = ∠ACB   ...(2) (Corresponding angles)

⇒ ∠ADE = ∠ACB   ...(3) (Given)

From equations (i), (ii) and (iii),

∠ABC = ∠ACB

⇒ AB = AC

Hence, ΔABC is an isosceles triangle.

Hence Proved.