# The Solution of Menaechmus

- Author:
- Ku, Yin Bon (Albert)

**Menaechmus**(380 - 320 BC) was a Greek mathematician. In fact, he was the mathematics tutor of

**Alexander the Great**. His most famous discovery was on

**conic sections**- He was the first to show that the three special curves -

**parabola**,

**hyperbola**and

**ellipse**- are obtained by cutting a cone in a plane not parallel to the base. He gave two solutions to double mean proportionals problem. Both involve the intersection of special curves. Given any two positive real numbers and , we want to find and such that

.

**First solution**: We construct a parabola and a hyperbola (the blue and black curves in the diagram below) that satisfy the following equations respectively:

The intersection point of these two curves is . Its coordinates are the required values of and .

**Second solution**:We construct two parabolas (the blue and red ones in the diagram below) that satisfy the following equations respectively:

The intersection of these two curves is . Its coordinates are the required values of and .