here. First the pencil consists of two parts. Hyperbolas and ellipses. There are also three singular or degenerate

members and a parabola. The conic is determined by the location of point P [SteinerVor, II, p. 231].

The parabola (blue) occurs when P is at the middle T of EF or any other than {A,B,C,D} point of the parabola.

All points P on the same half-plane of DC with E and outside the parabola define hyperbolas. The same happens

when P is inside the parabola (its convex region) and between the parallels AB, CD. The same happens also when

P is outside the parabola and below the line AB.

Ellipses are obtained when P is on the upper plane of DC and inside the parabola. Also when P is in

the parallel region and outside the parabola and also in the lower region of AB and inside the parabola.

The centers of all conics of the pencil are on line GG' since obviously this is a diameter for all of them.

The singular conics are three in number and represented by the union of lines a) {AD,BC}, b) {BD,AC} and

the parallels {AB, CD}.

For the hyperbolas of the pencil look also at PowerGeneralHyperbola.html and PowerGeneralHyperbola2.html .

For the parabola look at ParabolaCircumscribingTrapezium.html .

[2] Find the rectangular hyperbola circumscribing a trapezium.

PowerGeneralHyperbola2.html

ParabolaCircumscribingTrapezium.html

[Horwitz] Allan Horwitz

[SteinerVor] Vorlesungen ueber synthetische Geometrie, vols. I, II, Teubner Leipzig 1867

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