Question

X and Y are center of circles which intersect at A and C. XA and XC are produced to meet at B and D. Prove AB = CD.

Answer 1

On the diagram, construct lines BY, XY and DY.

Now, consider triangles ΔXBY and ΔXDY. We can say that these triangles are congruent by the SAS postulate (they share side XY, XY bisects ∠BYD so ∠BYX=∠DYX, and BY=DY=radius). If they're congruent, their sides are the same so BX=DX.

BX=DX

AB+radius = CD +radius

AB=CD