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Ptolemy’s theorem

  1. Oct 28, 2014 #1
    1. The problem statement, all variables and given/known data
    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.

    2. Relevant 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)

    3. 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

    upload_2014-10-29_4-21-49.png
     
  2. jcsd
  3. Oct 30, 2014 #2

    RUber

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    Homework Helper

    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.
     
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