let "c′" be the curve obtained by shrinking "c" by a factor of 2 toward "p". then the point "r′" corresponding to "r" is the center of the rectangle "pxry", and the tangent to "c′" at "r′" bisects this rectangle parallel to "py" and "xr". a ray of light starting from "p" and reflected by "c′" at "r' "will then pass through "y". the reflected ray, when extended, is the line "xy" which is perpendicular to the pedal of "c". the envelope of lines perpendicular to the pedal is then the envelope of reflected rays or the catacaustic of "c′". this proves that the catacaustic of a curve is the evolute of its orthotomic.