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### 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)

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

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

-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

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