Similarity: Dilations and Cones
The construction of a dilation is closely related to the "eyeball test" for similarity: Hold up two plane figures in parallel planes, close one eye, and move the figures until they align exactly or you know that they can't.
The first graphics view shows a large triangle, ABC.
The 3D graphics view shows the eyeball test with a smaller, similar triangle, A'B'C'. Lines of sight show A and A' lined up with the eye; the same holds for B and B and C and C'.
By clicking the check boxes in the first graphics view, you can see a parallel projection of this 3D figure into the plane of ABC. The flattened pictures is the projection, which is also the dilation construction.
The third checkbox, Show plane EAB, shows this plane in the 3D window. The right hand graphics view shows this plane in 2 dimensions.
For you to do: use Euclid's definition of similar polygons (corresponding angles equal, corresponding sides proportional) to prove that
(a) a dilation produces a similar figure and
(b) the eyeball test really does determine similar figures.