Since point is the image of point after a dilation with center and scale factor Likewise, if any point on the smaller circle is dilated through by a factor of its image will fall on the larger circle. Hence, the circles are related by a dilation, and so, are similar. The same argument can be used for any two concentric circles. This leads to the following conclusion.
Concentric circles are similar.
Dilate the concentric circles below so that they map onto each other.
Note that the circles have different radii. The radius of the smaller circle is and the radius of the larger circle is
It can be seen in the diagram that the radius of the smaller circle is and the radius of the larger circle is
Non-concentric circles are similar.
With this information and knowing that any two concentric circles are also similar, a more general statement can be made.
All circles are similar.
It can be seen in the diagram that the radius of both circles is
It can be seen in the diagram that the smaller circle has a radius of and its center at the point Furthermore the larger circle is centered at and its radius is
In this lesson, it has been proven that all circles are similar by using similarity transformations. Therefore, any theorem that is valid for one circle, is also valid for all circles. This can be exemplified by the following theorem.