SOLUTION: If D , E and F are the midpoints of sides AB , BC , and CA respectively of an equilateral triangle ABC , prove that triangle DEF is itself an equilateral triangle.

Algebra ->  Geometry-proofs -> SOLUTION: If D , E and F are the midpoints of sides AB , BC , and CA respectively of an equilateral triangle ABC , prove that triangle DEF is itself an equilateral triangle.      Log On


   

Question 490853: If D , E and F are the midpoints of sides AB , BC , and CA respectively of an equilateral triangle ABC , prove that triangle DEF is itself an equilateral triangle.
Answer by richard1234(7193) About Me  (Show Source):
You can put this solution on YOUR website!
You will see that, upon drawing D,E,F and forming DEF, that there are three "outer" triangles, and triangle DEF. Each of these "outer" triangles has a 60 degree angle, and its two adjacent sides are 1/2 the length of a side length of ABC; hence the outer three triangles are equilateral. This implies DEF is also equilateral, because the side lengths of DEF are the side lengths of the other three triangles.