The segment formed by joining the incenter and centroid of a certain right triangle is parallel to one of its legs and has a length of 1 unit.
Find the perimeter of this triangle.
Let AHB be the right triangle with a, h, and b
the lengths of sides HB, BA, and AH
respectively ( h^2 = a^2 + b^2 ). Let I denote
the incenter and M the centroid. We will
assume that b > a. Let r denote the inradius
with r = ( a + b  h ) / 2.
IM parallel to AH ==> a/3 = r
==> a = 3b/4 & h = 5b/4
IM = b/3  r ==> IM = b/12
Perimeter = ( a + b + h ) / IM
= ( 3b/4 + b + 5b/4 ) / b/12
= 36
Edited on March 4, 2019, 11:51 pm

Posted by Bractals
on 20190304 23:47:55 