HyperbolaGeneratedCircleAndPoint

Hyperbola Generation

Point H moves on circle c = E(EK) and L is the second intersection of the line from fixed B to H with the circle. Project on line EL point B parallel to EH.Then this projection O lies on a hyperbola,
whose center is point I, the intersection of BE with the circle nearest to B.The asymptotes of the hyperbola build an angle equal to the angle viewing the circle from B.

- A key feature is the isosceles BOL, which implies the hyperbola property with foci at B, E.
- J, N are respectively the middles of BI and BK.
- The statement on the angle of asymptotes follows from the consideration of the limit positions of H, when H is identical with G.
In this case OE is orthogonal to BG and by the isosceles BO is parallel to it. Hence one asymptote from I is parallel to EG.
Analogous is the reasoning for the other asymptote.
The figure is build starting with an angle ABC i.e. by prefixing the angle of the asymptotes.
Thus all hyperbolas resulting by varying the location of the center E of the circle (so that it remains tangent to ABC) are homothetic to each other.
Additional properties:
- BIG is isosceles.
- BI = IG = IE and GN is orthogonal to BE.
- Analogously to the creation of O project B to S on EH parallel to EL. The S lies also on the same hyperbola.
- S is the symmetric of O with respect to the center I of the hyperbola.

Hyperbola Generation

See Also