If A, B, C, D are four points on a circle in order such that AB = CD, prove that AC = BD.

Question

If A, B, C, D are four points on a circle in order such that AB = CD. How do you prove that AC = BD.

Answer 1

$AB=CD$ and $AO=BO=CO=DO=R \implies$
$\triangle AOB \cong \triangle COD \implies$
$\angle AOB = \angle COD$

$\angle AOC = \angle BOD$ and $AO=OC=OB=OD\implies$
$\triangle AOC =\triangle BOD \implies$
$AC=BD$

Answer 2

Let $M$ be the midpoint of the arc $BC$. Then the complete figure is symmetric with respect to the line $O\vee M$.