2017 AMC 12B Problems/Problem 15
Let be an equilateral triangle. Extend side beyond to a point so that . Similarly, extend side beyond to a point so that , and extend side beyond to a point so that . What is the ratio of the area of to the area of ?
Let . Then, the area of the small (inside) equilateral triangle is . Therefore the denominator of the ratio must be .
Recall The Law of Cosines. Letting , . This simplifies to . Since both and are both equilateral triangles, they must be similar due to similarity. This means that .
Therefore, our answer is .