Assignment 4

Three Medians of a Triangle are Concurrent

by Emily Bradley

In a triangle ABC, prove that the three meidans, that is the lines connecting the vertices to the opposite midpoints, (AF, BE, CD), concur at point G.

Proof

Given	Triangle ABC with D, E the midpoints of AB and AC.
Proof	Let D be the midpoint of AB. Then DC is a median. Let E be the midpoint AC. Then EB is a median.Call the point where DC intersects with FB, G. Because AD = 1/2 AB and AE = 1/2 AC, and angle A is shared, Δ ADE = 1/2Δ ABC, so DE is parallel to BC.
	Now, ∠EDG≅ ∠GCB, since they are alternate angles interior angles between parallel lines DE and BC with transversal DC. Also, ∠DEG ≅ ∠CBG, since they are alternate angles interior angles between parallel lines DE and BC with transversal EB. Also BC = 2DE, because Δ ADE = 1/2Δ ABC. Therefore Δ BGC = 1/2Δ EGD. Particularly BG ≅ 2GE. Thus the point G lies on the median BE 2/3 of the way from the vertex B to the opposite side's midpoint E.
	Reverse the roles of A and B. We have a third median EB that also passes through G. Thus all three medidans concur in the point G.