The locus of D is the (red) eye curve shown.

From its definition follows that:

1) Every secant throught A intersects the curve at two points {D,F}.

2) The same secant intersects the circle at {B,C} and |DB|=|FC|.

3) The common to DF and BC middle E describes a circle with diameter AO.

4) The ratio ED/EB is constant and equal to m = (k+1)/(k-1).

5) The curve is symmetric on the axis OA.

6) Setting OA = a, R for the radius of the circle the polar equation with center at A is:

(r-acos(fi))

7) In cartesian coordinates centered at A and x-axis AO:

(x

8) Inverting this curve with respect to the circle of radius L=|AH|, centered at A we obtain an ellipse.

In fact, the inverted radius is r' = L

The figure displays the ellipse obtained through the inversion with radius L = |AH|. It is characterized by its

properties:

a) To be tangent to {AH, AI} respectively at points {H, I} and,

b) to pass through K', which is the inverse of K,

c) its focal points are on the circle through {H,I,A}.

See Inversion.html , Ellipse.html .

Ellipse.html

Produced with EucliDraw© |