SimilarInscribed

1. Similar shapes inscribable in a circle

Polygons (here triangles CDE and CAG) similar to each other w.r. to a common vertex (here C) and
inscribable in circles have the lines joining homologous vertices pass through the second intersection point of
their circumcircles (here point O).

The key fact is that triangle (DCA) is similar to (CEG). The rotation-angle involved in the similarity about
the center C is equal to ang(ECG), by which also line CE is rotated so as to take a position along line CG.
Thus the angle GCA, by which triangle DCA rotates to obtain aposition such that its sides {CD,CA} go
along the sides {CE, CG} is equal to the angle of lines DA and EG, intersecting at a point O. Thus, GCOA
is a cyclic quadrangle.

2. Similar shapes inscribable in a circle (general case)

The figure below illustrates the general case of two similar cyclic polygons, the similarity being centered
at the common vertex A of the two polygons. All other homologous vertices of the two polygons define
lines passing through the other intersection point B of the two circumcircles.
The argument for the general case is the same with the previous one for the case of triangles.

This key-fact on the circumcircles was at the basis of the discussion in the file SimilarlyGliding.html as
well as in the file Similarity.html , handling the foundamentals on Similarities.

1. Similar shapes inscribable in a circle

2. Similar shapes inscribable in a circle (general case)

See Also