The perimeter of a triangle, its area, and the radius of the circle incsribed 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.
The perimeter of a triangle, its area, and the radius of the circle incsribed 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.
the incenter of a triangle is equidistant from each side of the triangle, that perpendicular distance being the radius, r, of the inscribed circle.
connect each vertex of the triangle to the incenter.
three smaller triangles are formed ... all with height r and base equal to one side of the large triangle.
let the sides of the triangle be a, b, and c
area of the triangle, A = (1/2)ra + (1/2)rb + (1/2)rc = (r/2)(a + b + c)
2A = r(a + b + c)