Triangle question based on the medians and angle bisectors

Question

A triangle has three sides which are consecutive integers. One of its medians is perpendicular to one of its angle bisectors. What is the perimeter of the triangle. So first i equated all the sides to x,x+1,x+2. I tried to find some relation between sides and medians but there seems to be too many unknown values. Please suggest any approach i should take. Thanks

Answer 1

This is a tricky one :)

$BM$ is the median and is perpendicular to $CP$. This means that $CP$ is really the height of the triangle $MCB$. From construction, the height splits the angle in $C$ exactly in half, which means that $MCB$ is an isosceles triangle, in other words that $BC = CM$.

To finish this off, $BC = CM = AC / 2$ and using that the lengths are consecutive integers the only solutions are:

• $ (AC, CB, BA) = (4, 2, 3)$
• $ (AC, CB, BA) = (2, 1, 3)$

The last one is impossible as the lengths don't obey the triangle inequality, so the solution is $(4,2,3)$.