# Proof: An Isosceles Triangle inscribed in a Circle

• Sep 6th 2011, 12:21 AM
goby
Proof: An Isosceles Triangle inscribed in a Circle
http://img69.imageshack.us/img69/5044/imagemta.jpg
Hypothesis: AC = CB
DE is the diameter
CH is perpendicular to AB (the altitude)
Thesis: DE : BC = BC : CH (BC^2 = DE * CH)
• Sep 6th 2011, 12:46 AM
alexmahone
Re: Proof: An Isosceles Triangle inscribed in a Circle
Quote:

Originally Posted by goby
http://img69.imageshack.us/img69/5044/imagemta.jpg
Hypothesis: AC = CB
DE is the diameter
CH is perpendicular to AB (the altitude)
Thesis: DE : BC = BC : CH (BC^2 = DE * CH)

$A=\frac{1}{2} *AB*CH$

$R=\frac{AB*BC*CA}{4A}$ (Circumradius - AoPSWiki)

$R=\frac{AB*BC*CA}{4*\frac{1}{2}*AB*CH}$

$R=\frac{BC*CA}{2CH}$

$R=\frac{BC^2}{2CH}$

$\frac{BC^2}{CH}=2R=DE$

$BC^2=DE*CH$
• Sep 6th 2011, 01:19 AM
goby
Re: Proof: An Isosceles Triangle inscribed in a Circle
Okay, I understood it, but what's the proof for R = http://mathworld.wolfram.com/images/...s/Inline22.gif?
• Sep 6th 2011, 01:25 AM
alexmahone
Re: Proof: An Isosceles Triangle inscribed in a Circle
Quote:

Originally Posted by goby
Okay, I understood it, but what's the proof for R = http://mathworld.wolfram.com/images/...s/Inline22.gif?

Area of a Triangle in Terms of Circumradius - ProofWiki