How to prove that certain points lie on the particular sides of a triangle?

Question

In triangle $ABC$ let $D,E,F$ be points on its sides such that

• $A$ and $D$ divide the perimeter of triangle into two equal parts
• $B$ and $E$ divide the perimeter of triangle into two equal parts
• $C$ and $F$ divide the perimeter of triangle into two equal parts

Prove that $D,E,F$ lie on $BC$, $AC$ and $AB$ respectively.

Answer 1

It suffices to prove that $D$ lies on $BC$; $E$ and $F$ follow by symmetry.

The triangle inequality implies that no side occupies more than half of the perimeter. If $D$ was on $AB$, $AD$ would be half the perimeter and thus $AB$ would be more than half the perimeter, a contradiction. Similar reasoning applies to $D$ on $AC$, so $D$ is on $BC$.