Given an oriented line
and a fixed point
on it, consider all trapezoids
one of whose bases
lies on
, in the positive direction. Let
be the midpoints of
and
respectively. Find the loci of vertices
of trapezoids that satisfy the following:
(i)
(
fixed);
(ii)
(
fixed);
(iii) the sum of squares of the nonparallel sides of the trapezoid is constant.
Remark
Remark. The constants are chosen so that such trapezoids exist.
![\Delta](/media/m/0/4/c/04cb29ce5ad4394d4492570217b5b0b2.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![\Delta](/media/m/0/4/c/04cb29ce5ad4394d4492570217b5b0b2.png)
![E,F](/media/m/e/d/9/ed9b3965cc5e0b074cdb261c465e4fe2.png)
![AB](/media/m/5/2/9/5298bd9e7bc202ac21c423e51da3758e.png)
![CD](/media/m/8/9/5/895081147290365ccae028796608097d.png)
![B,C,D](/media/m/f/5/b/f5b0be1be461521910aaefcb9a8196cb.png)
(i)
![|AB| \leq a](/media/m/b/1/1/b11fe86057eea0d97a229ccf65387010.png)
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
(ii)
![|EF| = l](/media/m/3/d/8/3d8455beb614c6e4cb29517326198e1c.png)
![l](/media/m/e/e/9/ee975101080f37986f56baaf4c3cdcd2.png)
(iii) the sum of squares of the nonparallel sides of the trapezoid is constant.
Remark
Remark. The constants are chosen so that such trapezoids exist.