Go to Fun_Math Content Table Three Famous Problems Delian Problem

Menaechmus (circa 380 BC - 320 BC), used parabolas and rectangular hyperbola to solve the Delian problem.

If x, y are two mean proportional between straight lines a, b, a / x = x / y = y / b then x |

********** Menaechmus_Delian_desc.dwg** *******

**To create this drawing : **
** Load Menaechmus.lsp (load "Menaechmus")**

Then from command line, type ** Menaechmus **

AO is set to length a.

A point C is selected on y-axis.

Draw a semi-circle with AC as diameter.

This arc cuts x-axis at point D.

Then ODC, and OAD form similar triangles.

Therefore , OD^{2} = CO x OA, or x^{2} = a y

P(x,y) is a point on the parabola x^{2} = ay

Another form of parabola: y^{2} = b x

A similar argument is applicable for this type.

****************** parabola_x2_ay_desc.dwg** ***************

You can see the process in animation.

**To create this drawing : **
** Load Menaechmus_Delian.lsp (load "Menaechmus_Delian")**

Then from command line, type ** test_draw_x2_ay **

To output jpg files for animation, use the executable **auto_draw_x2_ay**.

****************** hyperbola_xy_ab_desc.dwg** ***************

You can see the process in animation .

The length of line segments AO and BO is set to a, and b, respectively.

Draw a vertical line CD through point B.

Draw a horizontal line CE through point A.

Select a point F on X-axis.

Drop line FG perpendicular to CE.

Draw a line from G to O, and extend it until it intersects CD at point H.

Draw a horizointal line from H.

This intersects a vertical line through F at point P

Two triangles OAG and HBO are similar.Then OA/AG = HB/OB or a/x = y/b

Therefore xy = ab

And point P (x,y) is the point on rectangular hyperbola : xy = ab

**To create this drawing and animation: **
** Load Menaechmus_Delian.lsp (load "Menaechmus_Delian")**

Then from command line, type ** test_draw_xy_ab **

To output jpg files for animation, use **auto_draw_xy_ab**
after setting up jpgfile and jpg_yes flag.

In the drawing

ON = (b/a)^{1/3} and OM = (b/a)^{2/3}

When a = 1, b = 2, ON = (2)^{1/3} and OM = (2)^{2/3}

**To create this drawing : **
** Load Menaechmus.lsp (load "Menaechmus")**

Then from command line, type ** Menaechmus_1 **

Note that both angles BMN and MNA are 90 degrees.(Why ?)

This observation lead the group of Plato's academy to invent the device to find the Delian problem,which will
be described in the next section.

************ Menaechmus_Delian_1.dwg** *********

**To create this drawing : **
** Load Menaechmus.lsp (load "Menaechmus")**

Then from command line, type ** Menaechmus_2 **

************ Menaechmus_Delian_2.dwg** *********

**To create this drawing : **
** Load Menaechmus.lsp (load "Menaechmus")**

Then from command line, type ** Menaechmus_3 **

************ Menaechmus_Delian_3.dwg** *********

Go to Fun_Math Content Table Three Famous Problems Delian Problem

All questions/suggestions should be sent to Takaya Iwamoto

Last Updated Nov 22, 2006

Copyright 2006 Takaya Iwamoto All rights reserved. .