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?

By the Pythagorean Theorem: a²+b²=c²
But:
a=x+r
b=y+r
c=x+y
so,
(x+r)²+(y+r)²=(x+y)²
(x²+2xr+r²)+(y²+2yr+r²)=(x²+2xy+y²)
2xr+2yr+2r²=2xy
xy=xr+yr+r²

Now the area A of the triangle is:
A=ab/2=(x+r)(y+r)/2
2A=xy+(xr+yr+r²)=2xy
xy=A

This is the same interesting fact about xy that Richard came up with in his first post, but it does not rely on knowing in advance that the area can be expressed as inradius times the perimeter.

Posted by TomM on 2005-03-01 10:49:53