AIME questions

The points (0,0), (a,11), and (b,37) are vertices of an equilateral triangle. Find the value of ab.

Consider the equilateral triangle in the complex plane. That is, (a, 11) represents
the point a+11i and (b, 37) represents the point b+37i. (a + 11i){(sqrt3i/2)+12}=b + 37i
Then b+37i is a 60◦ counterclockwise
rotation of a + 11i about the origin, or (a + 11i)(cosine 60◦)= b + 37i

Setting the real and imaginary parts equal:
b= a/2-(11sqrt3)/2
37= 11/2+(asqrt3)/2

Solving we get a=21sqrt3 b=5sqrt3

ab=315[i]

Great job!
Rectangle ABCD is given with AB=63 and BC=448 Points E and F lie on AB and BC respectively, such that AE=CF=94 The inscribed circle of triangle BEF is tangent to EF at point P and the inscribed circle of triangle DEF is tangent to EF at point Q Find Q