Draw line AD to a variable point D of the circle and at D draw the orthogonal line DE to AD.

- Line DE envelopes an ellipse.

- The ellipse has the circle as its auxiliary circle and touches it at the diametral points with line AO.

- The contact point E of line DE is the projection on it of the fourth harmonic G = F(A,C), where F the intersection of AB with DE.

- Analogous generation is valid also for hyperbolas. The only difference is on taking A to be outside the circle.

The proofs follow immediately from the discussion in Ellipse.html .

Line HU is taken to be orthogonal at H to AE.

Line HU envelopes an ellipse.

The auxiliary circle of the ellipse is homothetic to the circle B(BE) w.r. to A and in ratio k to it.

The ellipses resulting for various k are all homothetic to each other.

The contact point U of line HU with the ellipse is the projection on HU of T, which is the harmonic fourth of K(A,S), where K is the intersection of HU with the line of centers AB.

The figure here is simply a homothetic image of the previous figure.

HyperbolaAsEnvelope.html

Produced with EucliDraw© |