Why does this work?
By: Tim Lehman

Given: Triangle ABC with D, E, and F as the midpoints of sides. Click here for a GSP sketch.

We want to see that all four triangles (BFD, DEC, FEA, and DEF) have the same area. To do this, we will see that each triangle has an area 1/4 as large as triangle ABC.

By definition of midpoint, BF=1/2BA and BD=1/2BC. Further, angle FBD=angle ABC because the angles share the same vertex and rays or sides.

Thus, triangle FBD~triangle ABC with a scale factor of 1/2. So, triangle FBD has 1/2 of the height and base of triangle ABC. Let h and b be the height and base of triangle ABC, respectively. Then, area (FBD) = 1/2 (1/2b*1/2h) = 1/4 (1/2bh). Therefore, triangle FBD has 1/4 of the area of triangle ABC. Similarly, triangle AFE and triangle EDC have 1/4 of the area of triangle ABC.

Because triangles FBD, EDC, and AFE are each 1/4 the area of ABC, then they together make up 3/4 of the area of the larger triangle. Thus, the remaining part of triangle ABC, triangle DEF, must be 1/4 of the area of triangle ABC as well. It then follows are four triangles have the same area so any combination of two to them would be equal in are to 1/2 of triangle ABC.

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