Answer to Question #160085 in Geometry for Kelvin Osei

Question

Prove that the line joining the midpoint of a median to a vertex of the triangle trisects

the side opposite the vertex considered.

Answer 1

Consider triangle ABC in which AJ is the median to BC and K is the midpoint of AJ while BK produced meets CA in P. We need to prove that CP/PA=2/1. Consider triangle AJC with BKP as the transversal. Apply Menelaus’s Theorem to triangle AJC by which, (AK/KJ)(JB/BC)(CP/PA)=1 (Ignoring the - sign since we know that one intersection is external). Therefore, (1/1)(1/2)(CP/PA)=1 or CP/PA=2/1; hence proven.