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Broken Hypotenuse (Posted on 2005-02-28) Difficulty: 3 of 5
Consider a right triangle with an inscribed circle. Let x and y be the lengths of the two line segments formed on the hypotenuse by the point of tangency with the circle. What interesting fact can you prove about x*y?

See The Solution Submitted by owl    
Rating: 3.6000 (5 votes)

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re: A possible solution (same result different way) | Comment 4 of 10 |
(In reply to A possible solution by Old Original Oskar!)

Call the radius of the circle R.

Construct the 3 radii to the points of tangency.  The triangle is disected into 2 kites and a square. 

The area by these pieces A= xr + yr + r^2
The area by the original triangle A= .5(x+r)(y+r)

Simplify xr + yr + r^2 = .5(x+r)(y+r)

2xr + 2yr + 2r^2 = xy + xr + yr + r^2

xy = xr + yr + r^2


  Posted by Jer on 2005-02-28 19:03:11
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