Alicia Conklin
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Proof Through Geometry

Picture
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

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

Picture
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.

 

Hint 2: Other Than a Triangle

Picture
Notice the parallelogram and use it to find more equal lines.
Therefore AE = DE = BF and EC = EF = BD.

Hint 3: Hint 3: Start with a
in a+d+f

Picture
Substitute a in for the line segments it represents from the triangle.
a = AE + EC

Hint 4
Hint 5
Solution
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