Question

Equilateral triangles ABX and ACY are described on sides AB, AC of a triangle ABC externally to triangle ABC. Prove that CX=BY.

Answer 1

Look at the diagram
u can see by cosine law
CX²=BC²+BX^2-2(BC)(BX)cos(60)
CX²=BC²+AB^2-(BC)(AB)...(1)
similarly
BY²=BC²+CY²-2(BC)(CY)cos(60)
BY²=BC²+AC²-(BC)(AC)
as AB=AC
BY²=BC²+AB^2-(BC)(AB)..(2)
(1)=(2)
therefore
BY=CY

Answer 2

in eqilateral triangle all the sides are equal . measure angle ABX=60 thus ab=bx=ax=60
thus it is an eqilateral triangle and as per given data abx=acy thus cx =by