Question

In the following figure, ∠AXY = ∠AYX. If `(BX)/(AX) = (CY)/(AY)`, show that triangle ABC is isosceles.

Answer 1

In the given figure,

∠AXY = ∠AYX

And `(BX)/(AX) = (CY)/(AY)`

To prove: ΔABC is an isosceles triangle

In ΔAXY

∠AXY = ∠AYX   ...(Given)

∴ AY = AX  ...(Sides opposite to equal angles)

`(BX)/(AX) = (CY)/(AY) => (AX)/(BX) = (AY)/(CY)`

∴ XY || BC

∴ ∠B = ∠AXY and ∠C = ∠AYX   ...(Corresponding angles)

But ∠AXY = ∠AYX is given

∴ ∠B = ∠C

∴ AC = AB  ...(Side opposite to equal angles)

∴ ΔABC is an isosceles triangle.