Max_Trap

Maximal/Minimal trapezium

Given a point O inside a circle c, consider all chords through O at a fixed angle ω. Show that the maximum/minimum quadrangles having diameters these chords are the trapezia having this angle for their diagonals. Show also that their symmetry axis is the line PO through the center of the circle.
Introduce the coordinate system shown below, where the circle centers at the x-axis at a point P(p,0) and the point O through which pass the chords at angle ω is the origin. The circle's equation is:
(1) (x-p)² + y² = R².
Its intersections with the parametric line through the origin:

To complete the discussion some details are needed, such as the case ω=π/2, the fact that there are no other solutions, the investigation of which values give maxima and which minima etc.. All these are left as exercises.
The analogous question for ellipses leads to a drudgery of computations.
The figure above shows the two extremal quadrilaterals which are trapezoids with their parallel sides orthogonal to line OP. The figures shows also another fact which, possibly, could be used to give a more geometric solution:
the pairs of opposite sides of the variable quadrangle ABCD intersect on the polar of O (see CyclicProjective.html ).
Remark The problem handled here is a special case of the following more general one:
Given are a conic (c) and a point O not on c. Let on the pencil O* of lines through O be defined a homography L'=F(L). For each pair of corresponding lines (L,L') define the corresponding quadrangle q(L) inscribed in c and having as diameters the lines (L,L'). Find the quadrangle q(L₀) with minimum/maxim area.

Maximal/Minimal trapezium

See Also