### Geometry of the Isosceles Trapezium and the Contra-parallelogram, with applications to the geometry of the Ellipse

R. F. Muirhead
1901 Proceedings of the Edinburgh Mathematical Society
In Figure 11 let ABCD be a contraparallelogram having AB = CD, AD = BC. Let O, P, Q, R be the midpoints of AB, BC, AD, DO respectively. They obviously lie in the line parallel to AC and to BD, and equidistant from them. Let U be the mid-point of AC and V that of BD. OURV is a rhombus each of whose sides is half of AD or BC, and parallel to AD or BC. PUQV is a rhombus each of whose sides is half of AB or CD, and parallel to AB or CD. and OP = AU = UC = QR. Let X be the point of intersection of
more » ... f intersection of AD and BC, and Y that of AB and CD. Then X, Y, U, V are collinear. Let M be the intersection of UV and OR. Let BP = aand OB = c. Then a 1 -r = B P -OB 2 = OV 2 -PV-= OM 2 -PM--O P . O Q = AU.BV. Now it is clear that a linkage formed of the jointed contraparallelogram ABCD and the two rhombuses OVRU, PVQU, jointed at their common points, will have one internal freedom of motion, and that if AB be fixed every point not in AB will describe a definite curve, which will be a circle except for points on CD.