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Triangle Circle (Posted on 2004-11-20) Difficulty: 3 of 5
Begin with a right triangle with hypotenuse h.

Inscribe a circle and label its radius, r.

What is the ratio of the area of the circle to that of the triangle?

See The Solution Submitted by SilverKnight    
Rating: 3.0000 (4 votes)

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Some Thoughts Thoughts, a method but not the answer | Comment 2 of 8 |
Easy part:  ratio is Pi*r^2 / (ab/2)
Hard part:  figuring out r as a function of a and b (and/or h)

So I'm thinking the answer has to be in terms of a function of a,b,h  but not r.

If the triangle is in the first quadrant with vertices at (0,0), (a,0), and (0,b), then the center of the circle (point O) should be at (r,r).  And the distance from point O to the hypotenuse is also r.

Unfortunately, I can't remember any of those relationships that help me figure out what r is, but I think I have 3 equations and 3 unknowns that should work.

Consider the point (x,y) which is where the circle touches the hypotenuse.  This point will satisfy 3 conditions:  it is on the hypotenuse, its distance from point O is r, and it is also on a line running through point O and perpendicular to the hypotenuse.

So here are the 3 equations:
Equation of hypotenuse line:  y=(-b/a)x + b
Equation of perpendicular line:  (y-r)/(x-r) = (a/b)
Distance to point O is equal to r:   (x-r)^2 + (y-r)^2 = r^2

And the 3 unknowns are x, y, and r
Should be able to eliminate x and y, and get r in terms of a and b.
But I haven't gotten that far yet.  Need more coffee.
  Posted by Larry on 2004-11-20 19:04:59
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