The Evolute of a Cardioid
Given a cardioid (red in the applets below), we can draw the osculating circle at each point. This circle has the same curvature as the cardioid. The centers of all those osculating circles lie on another cardioid (blue in the applets below) that is one-third the size. That is, the evolute of a cardioid is another cardioid. This fact implies that if we take a normal line to the original cardioid (a line that is perpendicular to the tangent line at that point), it is tangent to the smaller cardioid.
In the second applet, we see that if we draw enough of those normal lines, the underlying cardioid emerges as the envelope of the normal lines.