Problem:
In △ABC medians AD and BE intersect at G and △AGE is equilateral. Then cos(C) can be written as nmp, where m and n are relatively prime positive integers and p is a positive integer not divisible by the square of any prime. What is m+n+p ?
Answer Choices:
A. 44
B. 48
C. 52
D. 56
E. 60 Solution:
Because △GEA is equilateral and E is the midpoint of AC, it follows that △CEG is isosceles with vertex angle 120∘, so ∠CGE=30∘ and therefore ∠CGD=180∘−60∘−30∘=90∘. Without loss of generality, let equilateral triangle △AGE have side length 2 . Then AC=4, and because G is the centroid of △ABC, it follows that GD=21AG=1 and AD=3, as shown.
Applying the Pythagorean Theorem to △AGC gives GC=42−22=12. Then applying the Pythagorean Theorem to △CGD gives CD=12+12=13. Finally, applying the Law of Cosines to △ACD gives
cosC=2⋅4⋅1342+13−32=26513.
The requested sum is 5+26+13=(A)44.
OR
As in the first solution, let AG=2 and conclude that AC=4 and AD=3. Applying the Law of Cosines to △CAD gives
