Pappus' trisection using a conchoid

This demonstrates Pappus trisection of an angle, using a neusis construction. The angle to be trisected is the angle

between

and the positive

-axis. Consider the rectangle

whose diagonal opposites are

and

, and extend the line

. Draw a line from

to cross

and meet this new line at

in such a way that the distance

is 2. (This is the neusis construction.) This could be implemented with a ruler containing two marks a distance 2 apart, a putting the ruler through

so that one mark lies on the line

and the other mark lies on the line through

. The the angles

. To implement a neusis construction in GeoGebra, we can use a curve known as a conchoid. Given a point

, a line

, and a distance

, a conchoid is the locus of all points

for which the distance

, where

is the point at which the line

intersects

. In the example here,

(the origin),

is vertical line through

, and

. You can check that for any line through the origin, the distance between its intersections with

and

is 2. If

, say, then it is not hard to show that the polar equation of the conchoid is

which can be given parametrically as

So, given a point

on the unit circle, we obtain its dstance

from the

-axis, and then draw the appropriate conchoid. The line which trisects

is the line between the origin and the point at which the line through

intersects the conchoid.

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