[alogo] Hippopede Generalization

The Hippopede of Proclus has been defined in Hippopede.html . The figure below illustrates a case
of a variable isosceles trapezium resulting slightly more general than a corresponding trapezium in the
aforementioned reference. This is done as follows:
1) A variable chord BC of the fixed circle c turns about point A lying inside the circle.
2) From another fixed point O a parallel to BC is drawn and on it an isosceles BCDE is constructed such that
    the ratio of the bases DE/BC is a given constant s.
3) The vertices {D, E} vary then on a quartic which is the inverse of a conic with respect to O.

[0_0] [0_1] [0_2]
[1_0] [1_1] [1_2]
[2_0] [2_1] [2_2]
[3_0] [3_1] [3_2]

The figure shows also the locus of the intersections F of the lateral sides of the trapezium. The locus of F is
a circle c' connected to the triangle AQO and the ratio s. Thus c' is independent of the circle c. Its center G is
on line MN and such that GN/GM=s, where {M,N} the middles respectively of {QA, QO}. c' passes also
through Q (the center of c) and its projection P on AO. These facts can be easily deduced from the fact that
the middles {K,L} respectively of {DE, BC} move on circles with diameter {QO, QA} and the fact that the
ratio FK/FL = s is constant.
The claims on the top, about the inversion and the quartic can be proved by the methods discussed in the interesting
paper of Alperin on pedals of conics [Alperin].

See Also



[Alperin] Roger C. Alperin A Grand Tour of Pedals of Conics Forum Geometricorum, 4(2004)

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