Ptolemy and Cyclic Quadrilaterals

Prove that the product of the diagonals is the sum of the product of the pairs of opposite sides
Proof:
1. Explain the construction[br]2. Prove that the triangles in the first pair are similar. [br]3. Prove that the triangles in the second pair are similar.[br]4. Write out the proportions deduced from the above steps [br]5. Finish the proof[br](solution at the bottom of this page)
And what does this theorem say when the quadrilateral is a rectangle?
spoilers:
[math]\frac{AD}{EC}=\frac{BD}{BC}[/math] and therefore [math]AD\cdot BC=BD\cdot EC[/math][br][br][math]\frac{AE}{CD}=\frac{AB}{BD}[/math] and therefore [math]AB\cdot CD=AE\cdot BD[/math][br][br]adding both equations we get [math]AB\cdot CD+AD\cdot BC=\left(AE+EC\right)BD=AC\cdot BD[/math]
– Esther Gruenhut

Information: Ptolemy and Cyclic Quadrilaterals