Thread: Area of Triangles + quadrilaterals problem

Problem: Draw two of the medians of a triangle. This subdivides the interior of the triangle into four pieces: the 3 triangles and a quadrilateral. Show that two of the three small triangles hae equal area and that the area of the third is equal to that of the quadrilateral.

Attempt:
So its really hard to describe geometric diagrams. I will try:
I have Triangle ABC. I drew median from vertex C to its opposite side and median from vertex A to opposite side. The quadrilateral mentioned is in the corner that vertex B is. From vertex B I draw a line that goes to the intersection of the previous two medians. Now I have 4 little triangles and I know that the ones that have equal bases have equal areas (because they share a height). However I cannot prove more than this.

help??

Originally Posted by mulaosmanovicben

Problem: Draw two of the medians of a triangle. This subdivides the interior of the triangle into four pieces: the 3 triangles and a quadrilateral. Show that two of the three small triangles have equal area and that the area of the third is equal to that of the quadrilateral.

Suppose that the medians AD and CF meet at M. The two triangles ABD and FBC have the same area (because both of them have half the area of the whole triangle ABC). Now subtract the quadrilateral FMDB from each of those two triangles, and you are left with the triangles AFM and MDC, which must therefore have the same area.

Use a similar idea to prove the other part of the problem.

It might help to draw the third median