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

- Line DE envelopes a hyperbola.

- The hyperbola has the circle as its auxiliary circle and touches it at the diametral points with line AB. The foci are A and its symmetric w.r. to B.

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

- The asymptotes of the hyperbola form an angle equal to the angle of the tangents from point A.

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

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

See HyperbolaAsEnvelopeII.html for a generalization constructing hyperbolas homothetic to this one.

HyperbolaAsEnvelopeII.html

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