Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1994 based on Points of Equilateral triangle.

## Points of Equilateral triangles – AIME I, 1994

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

- is 107
- is 315
- is 840
- cannot be determined from the given information

**Key Concepts**

Integers

Complex Number

Equilateral Triangle

## Check the Answer

But try the problem first…

Answer: is 315.

AIME I, 1994, Question 8

Complex Numbers from A to Z by Titu Andreescue

## Try with Hints

First hint

Let points be on complex plane as b+37i, a+11i and origin.

Second Hint

then \((a+11i)cis60=(a+11i)(\frac{1}{2}+\frac{\sqrt{3}i}{2})\)=b+37i

Final Step

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.

## Other useful links

- https://www.cheenta.com/rational-number-and-integer-prmo-2019-question-9/
- https://www.youtube.com/watch?v=lBPFR9xequA

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