Why does a/SinA = b/SinB = c/SinC?

Date: 08/12/97 at 07:57:39
From: Glen Gibson
Subject: Trigonometry

I was just playing aroung with a proof of a triangle and came across 
the fact that a/SinA = b/SinB = c/SinC. Why?

Date: 08/12/97 at 08:47:05
From: Doctor Jerry
Subject: Re: Trigonometry

Hi Glen, 

Draw a triangle ABC, with AC horizontal and B above AC, maybe somewhat 
to the left of the center point of AC. This is so that we can be 
looking at the same triangle, more or less.  Draw a line from B 
perpendicular to and intersecting the line AC at E.  Let BE = h.  

Let a be the side opposite angle A, b opp. B, and c opp. C. 

Note that sin(A) = h/c and sin(B) = h/a.  Solve each equation for h 
and equate to get c*sin(A) = a*sin(C). 

This is part of the law of sines. You can finish by drawing another 
altitude.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/