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.

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].

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