**Proof Through Geometry**

The figure shows an equilateral triangle ABC with DE parallel to BC and EF parallel to AB. D and G are on AB; F and H are on BC. GM and HN are perpendicular to AC. If AC = a, DG = d, FH = f, and MN = x prove that x = (a + d + f) / 2

Equilateral Triangle ABC

AC = a

DE // BC, EF // AB

DG = d

FH = f

GM perpendicular to AC

HN perpendicular to AC

**Given:**Equilateral Triangle ABC

AC = a

DE // BC, EF // AB

DG = d

FH = f

GM perpendicular to AC

HN perpendicular to AC

**Prove:**

x=(a+d+f)/2## Hint 1: DE // BC, EF // AB

By having parallel sides with our large equilateral triangle, ABC, we notice that

triangle ADE and triangle EFC are also equilateral triangles. Therefore AD = DE = AE

and EF = FC = EC.

triangle ADE and triangle EFC are also equilateral triangles. Therefore AD = DE = AE

and EF = FC = EC.