Image:Ptolemy's theorem.svg

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Ptolemy's_theorem.svg‎ (SVG file, nominally 500 × 164 pixels, file size: 25 KB)

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[edit] Theorem

Diagram for geometric proof of en:Ptolemy's theorem. Created in w:Inkscape.

Let ABCD be a en:cyclic quadrilateral.
Note that on the en:chord BC, the en:inscribed angles ∠BAC = ∠BDC, and on AB, ∠ADB = ∠ACB.
Construct K on AC such that ∠ABK = ∠CBD;
1. Note that since ∠ABK + ∠CBK = ∠ABC = ∠CBD + ∠ABD, ∠CBK = ∠ABD.
Now, by common angles △ABK is en:similar to △DBC, and likewise △ABD ∼ △KBC.
Thus AK/AB = CD/BD, and CK/BC = DA/BD;
1. Thus AK·BD = AB·CD, and CK·BD = BC·DA;
2. Adding, (AK+CK)·BD = AB·CD + BC·DA;
3. But AK+CK = AC, so AC·BD = AB·CD + BC·DA; en:Q.E.D.

[edit] Summary

Description	Diagram of Ptolemy's theorem
Source	Self-made
Date
Author	This media is lacking author information. Česky \| Deutsch \| English \| עברית \| 한국어 \| Lietuvių \| Nederlands \| Polski \| Русский \| ‪中文(繁體)‬ \| +/-
Permission (Reusing this image)	see below

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