Show that in a right triangle the inradius circumradius and

Show that in a right triangle, the inradius, circumradius, and semiperimeter are related by the formula s = r + 2R.

Solution

Show that in a right triangle, the inradius, circumradius, and semiperimeter are related by the formula s = r + 2R

. Solution

Let c by the length of the hypotenuse, and a and b the lengths of the sides.

Then R = c/2,

s = (a + b + c)/2, and r = ab/(a + b + c)

Since c = a 2 + b 2, we get

r = ab (a + b + c) = ab(a + b c) (a + b) 2 c 2 = a + b c 2 = s c = s 2R.