In triangle ABC, D, E and F are the midpoints of BC, CA and AB, respectively.…

Concept

The segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Similar figures with side-length ratio k have area ratio k2.

Application

1. Because E and F are midpoints of CA and AB, EF is parallel to CB and EF/CB = ½.

2. Therefore, △AEF is similar to △ABC with side-length ratio ½, so Area(△AEF)/Area(△ABC) = (½)2 = ¼.

3. The three corner triangles △AEF, △BFD and △CDE are congruent; hence each has area ¼ Area(△ABC). The remaining medial triangle △DEF also has area ¼ Area(△ABC).

Cross-check

The midpoint segments partition △ABC into four congruent triangles, so their four equal areas add back to the area of △ABC.

Thus, Area(△DEF) = ¼ Area(△ABC).