Ptolemy's Theorem: Deriving 2sin(θ/2)

[PcumP_Ravenclaw]

Homework Statement

7. Show that if θ is real then $|e^{iθ} − 1| = 2\sin(\frac{θ} {2})$. Use this to derive
Ptolemy’s theorem: if the four vertices of a quadrilateral Q lie on a circle.
then $d1*d2 = l1*l3 + l2*l4$ where d1 and d2 are the lengths of the diagonals
of Q, and l1, l2, l3 and l4 are the lengths of its sides taken in this order around Q.

Homework Equations

using the identies, e^iθ = cosθ + i*sin(θ)
and cos^2(θ) + sin^2(θ) = 1
and cos(2θ) = 1 - 2sin^2(θ)

The Attempt at a Solution

using the identies I could show that $|e^{iθ} − 1| = 2\sin(\frac{θ} {2})$
but I am not sure about the derivation. I have drawn out the statements below

 

[RUber]

For this problem, I would start by writing the 4 points, A, B, C, D, as $re^{i\theta_A}, re^{i\theta_B}, re^{i\theta_C}, re^{i\theta_D}$. The distance between any 2 points is $d(re^{i\theta_1},re^{i\theta_2})=\sqrt{(r\cos\theta_1-r\cos\theta_2)^2+(r\sin\theta_1-r\sin\theta_2)^2}$
Using these distances, and the identities you already have, you should be able to verify the theorem.
I cannot see immediately where the equivalence you showed in part a is applicable, but I also have not worked all the way through this problem yet.