Drexel dragonThe Math ForumDonate to the Math Forum

Ask Dr. Math - Questions and Answers from our Archives
_____________________________________________
Associated Topics || Dr. Math Home || Search Dr. Math
_____________________________________________

Ptolemy's Theorem


Date: 09/07/97 at 08:28:13
From: Anonymous
Subject: Ptolemy's Theorem

Dear Dr. Math,

I am searching for the proof for the Ptolemy's Theorem. Can you please 
give me a reference? Thanks.

Almut Breitling


Date: 09/07/97 at 18:00:53
From: Doctor Anthony
Subject: Re: Ptolemy's Theorem

Ptolemy's Theorem
------------------

If a quadrilateral is cyclic the rectangle contained by its diagonals 
is equal to the sum of the rectangles contained by opposite sides.

Draw the circle with any cyclic quadrilateral ABCD.  Draw the 
diagonals AC and BD. Suppose that <BAC is greater than <CAD. In <BAC 
draw AP, meeting BD at P, so that <BAP = <CAD.

In triangles ABP, ACD, <BAP = <CAD (construction), and <ABP = <ACD 
(angles in same segment), so the triangles are similar and:

  AB     BP 
 ---- = ----    or  AB.DC = AC.BP   ......(1)
  AC     DC 


In triangles ABC, APD   <BAC = <PAD   (i.e. <BAP +<PAC = <CAD + <PAC)

also <ACB = <ADP    (angles in same segment).  Therefore triangles are 
similar, and

   BC     AC
  ---- = ----    or   BC.AD = AC.PD   ......(2)
   PD     AD

Combining (1) and (2)

   AB.DC + BC.AD = AC.BP + AC.PD

                 = AC(BP + PD)

                 = AC.BD

and so the theorem is proved.

If ABCD is not a cyclic quadrilateral, the theorem states

  AB.CD + AD.BC > AC.BD

Construct <BAP = <CAD  and  <ABP = <ACD

In this situation <ABD does not equal <ACD and therefore <ABP does not 
equal <ABD and P does not lie on BD.

Then BP + PD > BD  and

       AB.CD + AD.BC > AC.BD

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Geometry
High School Triangles and Other Polygons

Search the Dr. Math Library:


Find items containing (put spaces between keywords):
 
Click only once for faster results:

[ Choose "whole words" when searching for a word like age.]

all keywords, in any order at least one, that exact phrase
parts of words whole words

Submit your own question to Dr. Math

[Privacy Policy] [Terms of Use]

_____________________________________
Math Forum Home || Math Library || Quick Reference || Math Forum Search
_____________________________________

Ask Dr. MathTM
© 1994-2013 The Math Forum
http://mathforum.org/dr.math/