Let
be a convex quadrilateral whose vertices do not lie on a circle. Let
be a quadrangle such that
are the centers of the circumcircles of triangles
, and
. We write
. Let us define ![A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).](/media/m/0/f/2/0f29b408ae769252715c3d8ec5648ec3.png)
(a) Prove that
and
are similar.
(b) The ratio of similitude depends on the size of the angles of
. Determine this ratio.
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![A'B'C'D'](/media/m/e/8/1/e81445bdb3d123a9f3b5c7ffad7eed8f.png)
![A',B', C',D'](/media/m/3/e/0/3e0dbfc1d1b1fef340a8d6faa295dfa0.png)
![BCD,ACD,ABD](/media/m/3/5/2/35284ec87060129f99f41a7bb21425cf.png)
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
![T (ABCD) = A'B'C'D'](/media/m/5/c/9/5c9ee3e94a780a02ffb3dcb15d13b013.png)
![A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).](/media/m/0/f/2/0f29b408ae769252715c3d8ec5648ec3.png)
(a) Prove that
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)
![A''B''C''D''](/media/m/a/b/b/abb48bbd1b091c941818c50a5dbbf373.png)
(b) The ratio of similitude depends on the size of the angles of
![ABCD](/media/m/9/c/e/9ce25711ba18d9663b73c3580de4bf5a.png)