Problem:
Suppose that △ABC is an equilateral triangle of side length s, with the property that there is a unique point P inside the triangle such that AP=1,BP=3, and CP=2. What is s ?
Answer Choices:
A. 1+2
B. 7
C. 38
D. 5+5
E. 22 Solution:
We begin by rotating △APB counterclockwise by 60∘ about A, such that P↦Q and B↦C. We see that △APQ is equilateral with side length 1 , meaning that ∠APQ=60∘. We also see that △CPQ is a 30−60−90 right triangle, meaning that ∠CPQ=60∘. Thus, by adding the two together, we see that ∠APC=120∘.
