Question

abc is an isosceles triangle inscribed in a circle and AB=AC the bisector of angle B and angle C meet the the circumference at X and Y respectively prove that BY= AX​

Answer 1

Given :  abc is an isosceles triangle inscribed in a circle and AB=AC the bisector of angle B and angle C meet the the circumference at X and Y respectively

To find :  BY = AX

BX = CY

Solution:

ΔABC  is isosceles triangle

AB = AC

=> ∠B = ∠C

BX bisector of angle B

=> ∠ABX =   ∠B/2

CY is bisector of angle C

=> ∠BCY = ∠C/2

∠B = ∠C

=> ∠ABX  = ∠BCY  

chord subtending equal angles are congruent .

=> AX = BY

QED

Hence Proved

bisector of angle B and angle C meet the the circumference at X and Y respectively

=> ∠CBX  =  ∠B /2

  ∠BCY  =   ∠C /2

∠B = ∠C

=> ∠CBX  =  ∠BCY

Comparing Δ ∠CBX   &  ΔBCY

CB = BC   ( common)

∠CBX  =  ∠BCY

∠BXC =  ∠CYB    ( angle by same chord//arc BC )

=> Δ ∠CBX   ≅  ΔBCY    AAS criteria

=> BX = CY

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