Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.
The points (0,0), (a,11), and (b,37) are the vertices of equilateral triangle, find the value of ab.
Integers
Complex Number
Equilateral Triangle
Answer: is 315.
AIME I, 1994, Question 8
Complex Numbers from A to Z by Titu Andreescue
Let points be on complex plane as b+37i, a+11i and origin.
then \((a+11i)cis60=(a+11i)(\frac{1}{2}+\frac{\sqrt{3}i}{2})\)=b+37i
equating real parts b=\(\frac{a}{2}-\frac{11\sqrt{3}}{2}\) is first equation
equating imaginary parts 37=\(\frac{11}{2}+\frac{a\sqrt{3}i}{2}\) is second equation
solving both equations a=\(21\sqrt{3}\), b=\(5\sqrt{3}\)
ab=315.
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