geometry

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The perimeter of a triangle, its area, and the radius of the circle inscribed in the triangle are related in an interesting way. Prove that the radius of the circle times the perimeter of the triangle equals twice the area of the triangle.

  • geometry -

    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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