## 1. Circumconics of a trapezium

The figure below displays the pencil of circumconics of trapezium ABCD. There are various properties to notice
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. Problems

[1]  Find the minimal ellipse circumscribing a trapezium ([Chong], [Horwitz]).
[2]  Find the rectangular hyperbola circumscribing a trapezium.

PowerGeneralHyperbola.html
PowerGeneralHyperbola2.html
ParabolaCircumscribingTrapezium.html

### Bibliography

[Chong] F. Chong A minimal ellipse in an ionospheric problem Mathematical Gazette, vol. 50, no 372
[Horwitz] Allan Horwitz Ellipses of minimal area and of maximal eccentricity circumscribing a convex quadrilateral Preprint, 2007
[SteinerVor] Vorlesungen ueber synthetische Geometrie, vols. I, II, Teubner Leipzig 1867