# Medians of an Isosceles triangle

## Homework Statement

Prove that the medians to the equal sides of an isosceles triangle divide each other into respectively equal parts

## The Attempt at a Solution

suppose we have a triangle ABC where AB = AC. Let D be the point on AB in which the median intersects AB, and let E be the point on AC in which the other median intersects AC. Consider triangles ACD and ABE. We know AC = AB. Also AD = AE because the medians are bisecting two congruent lines. Also note that ∠CAD = ∠BAE. Therefore triangle ACD is congruent to triangle ABE by SAS. It follows that the medians CD and BE are congruent.

This is as far as i get. I can show that the medians are congruent, but I do not know how to show they divide each other into equal line segments

LCKurtz
Homework Helper
Gold Member

## Homework Statement

Prove that the medians to the equal sides of an isosceles triangle divide each other into respectively equal parts

## The Attempt at a Solution

suppose we have a triangle ABC where AB = AC. Let D be the point on AB in which the median intersects AB, and let E be the point on AC in which the other median intersects AC. Consider triangles ACD and ABE. We know AC = AB. Also AD = AE because the medians are bisecting two congruent lines. Also note that ∠CAD = ∠BAE. Therefore triangle ACD is congruent to triangle ABE by SAS. It follows that the medians CD and BE are congruent.

This is as far as i get. I can show that the medians are congruent, but I do not know how to show they divide each other into equal line segments

Draw DE. You should be able to show triangle BDE is congruent to triangle CDE and that DE is parallel to BC. If O is where the medians intersect, show triangle DEO is similar to triangle BCO. That should give you the proportional sides you seek.