[b][color=#0000ff]Menaechmus[/color][/b] (380 - 320 BC) was a Greek mathematician.  In fact, he was the mathematics tutor of [b][color=#0000ff]Alexander the Great[/color][/b].  His most famous discovery was on [b][color=#0000ff]conic sections[/color][/b] - He was the first to show that the three special curves - [b][color=#0000ff]parabola[/color][/b], [b][color=#0000ff]hyperbola[/color][/b] and [b][color=#0000ff]ellipse[/color][/b] - are obtained by cutting a cone in a plane not parallel to the base.[br][br]He gave two solutions to double mean proportionals problem.  Both involve the intersection of special curves.  Given any two positive real numbers [math]a[/math] and [math]b[/math], we want to find [math]x[/math] and [math]y[/math] such that [br][br][center][math]a:x=x:y=y:b[/math].[/center]  [br][br][b]First solution[/b]: We construct a parabola and a hyperbola (the blue and black curves in the diagram below) that satisfy the following equations respectively:[br][br][center][math]y^2=bx[/math][br][math]xy=ab[/math][/center][br]The intersection point of these two curves is [math](x,y)[/math].  Its coordinates are the required values of [math]x[/math] and [math]y[/math].[br][br][b]Second solution[/b]:We construct two parabolas (the blue and red ones in the diagram below) that satisfy the following equations respectively:[br][br][center][math]y^2=bx[/math][br][math]x^2=ay[/math][/center][br][br]The intersection of these two curves is [math](x,y)[/math].  Its coordinates are the required values of [math]x[/math] and [math]y[/math].