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: Create Midpoint k on a

Picture
There is now a line BK that  splits the equilateral triangle in half.
We now have two  30-60-90 triangles. With these triangles come
 special properties such as the hypotenus is twice as much as the base.
Therefore AB = 2AK and BC = 2KC

Hint 5: Mess With the x

Picture
Now x = MK + KN = GT + VH and still holding the same properties as the previos hint.
Therefore BG = 2GT and BH = 2VH. Use this information to substitute in the eaquarions and information gathered.


Solution

Picture
1. By looking at the triangles we can say that a = AE + EC

2. a  + d + f  = AE + EC + d  + f

3.  Replace AE with equivalent BF, and EC with DB and  you can
now measure two line segments.

4.  BF + DB + d  + f  = GB + BH

5.  Make midpoint K. By making this midpoint we have two 30-60-90 
triangles.

6.  We can draw two new line segments such that we're measuring
 x using these 30-60-90 triangles.

7. GB + BH = a + d + f

 8. x = MK + KN but we can now use our new values x = GT + VH

 9. GB = 2 GT  and  BH = 2VH

 10. Substituting values in for step 7 we get: a + d + f = 2GT + 2VH

 11. a  + d + f = 2(GT + VH)

12. a  + d + f = 2(x)

 13. Dividing both sides by 2: x = (a  + d + f) / 2


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