Trisecting an angle, using only compass and straight edge, was one of the great classical problems of antiquity.

Modern mathematics has proved it impossible, but here is a simple and ingenious mathematical cardboard device that trisects accurately:


If you place it properly, so the edge AB passes through the vertex of the given angle, one side of the angle passes through point C, and the other side tangents the arc MN (arc of the circumference centered at E and radius ED), the lines traced from the vertex through the points D and E trisect exactly the angle.

Prove it.

The device is perfectly drawn to scale, so you can get any information you need about lengths, parallelism, intersections etc...

From the configuration of the device we have
|CD| = |DE| = |EN| and edge AB coincides with
the perpendicular bisector of line segment CE.

If the vertex of the angle lies on edgr AB,
one side passes through point C, and the other
side is tangent to arc MN at point T; then
triangles ADC, ADE, and AET are congruent
right triangles.

Therefore, angle CVT is trisected.

 

Posted by Bractals on 2008-08-24 14:25:47