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December 22, 2014

December 22, 2014

Posted by **sarah A** on Sunday, April 11, 2010 at 9:48pm.

- geometry -
**Reiny**, Sunday, April 11, 2010 at 10:54pmConstruct "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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