Posted by sarah A on Sunday, April 11, 2010 at 9:48pm.
Construct "any" triangle. Do not use a special triangle such as an equilateral or right-angled triangle
the inscribed circle will have its centre at the intersection of the angles of the triangle.
Draw a radius to each of the sides, call it r.
It is trivial to see that we have 3 pairs of congruent triangles. Let's call their bases a, b, and c
So the perimeter of the triangle = 2a + 2b + 2c
So the product of the radius and the perimeter is
r(2a + 2b + 2c) = 2r(a+b+c)
Now the area of the whole triangle
= 2(1/2)ar + 2(1/2)br +2(1/2)cr
= ar+br+cr
= r(a+b+c)
All Done!
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