Trigonometry + Geometry

Date: 5/5/96 at 12:39:45
From: Anonymous
Subject: Trigonometry and Geometry
 
A circle is inscribed in a right angle triangle (90 degrees), and another 
circle is circumscribed around the triangle.  The ratio of the radii of 
the circles is 13 to 4.  

What are the other two angles of the triangle?

Date: 10/31/96 at 17:13:31
From: Doctor Ceeks
Subject: Re: Trigonometry and Geometry

Hi,

If R is the circumscribed circle radius and r is the inscribed circle
radius, and a,b,c are the sides of the triangle, we have these 
equations:

Pythagoras:  a^2 + b^2 = c^2

Computation of areas:  ab = (a+b+c)r

Hypotenuse of right triangle is diameter of circumscribed circle: 
c = 2R

Because angles don't change under dilation, you can assume that
r = 4, R = 13.

That gives three equations in three unknowns, and you can solve them
and ultimately compute arccos(a/c), arccos(b/c) for the answer.

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