X is a point inside an equilateral triangle ABC. On BX
another equilateral triangle is drawn, Y and A being on
the opposite side of BX. Prove that AX = CY.
Given :
ABC is equilateral triangle
BYX is equilateral triangle
To prove:
AX = CY
Proof:
In triangle ABC
AB=BC=CA
In triangle XBY
XB=BY=XY
Consider triangleABX and triangleCBY
AB=BC
BX=BY
Angle ABC=angle XBY=60 (EQUILATERAL TRIANGLES)
Subtract angleXBC on both sides
angleABX=angleCBY
Hence by sas congruence rule
triangle ABX is congruent to triangle CBY
AX=CY(BY CPCT)