Draw a circle with center A through B. Draw a point C and construct the inversion C' of C (in a way that works for all C). The construct the line c through C' perpendicular to C; this line c is the polar of C. Hide auxiliary lines so that the figure looks like one of these, depending on where C is located.

Use this figure that makes a script called polar.gss.

In a new sketch, again draw a circle with center A through B and draw a line CD. Label the line p. Let E be the foot of the perpendicular from A to p. Let P be the inversion of E.

Make a script CAREFULLY following these instructions. Hide the line AE and the point E. Select everything in the figure except points C and D. Then choose Make Script. The given in your script should be A and B and a straight object (not 4 points). Name this script pole.gss.

In a new sketch, draw a circle with center A through B. This will be the circle in which you construct the pole of a line. The line will be on the simplest "line conic". It will be a tangent line to a circle. So construct a second circle with center C through D. Construct E as a point on this circle and the line k as the perpendicular to CE. As you drag E the line k moves as a tangent to the circle.

Now construct the pole K of k. Then either trace K or make a locus of K to see what shape K traces at the line k moves around the circle.

In a new sketch, draw a circle with center A through B. This will be the circle in which you construct the pole of a point. The point will be on the simplest "point conic". It will be a point on a circle. So construct a second circle with center C through D. Construct E as a point on this circle.

Now construct the polar e of E. Then either trace e or make a locus of e to see what shape e traces at the point E moves around the circle.

This time draw two circles, c1 with center A through B and c2 with center C through D. Then draw a third circle and with center E through F. Let G be a point on this circle. Then construct the polar g of G in c1, followed by the pole H of g in c2. Trace the locus of H as G moves.

Draw a triangle ABC and construct the polars of A, B, C and the poles of the 3 sides. What does this look like?

Inscribe a circle in this triangle, then take the pole of a (movable) tangent to this circle to form an ellipse as a locus. What do you get?

Repeat the experiment with a circumscribed triangle.