structural:expansion, quality:correction
Proof

In a Cartesian coordinate system circle k {\displaystyle k} may have midpoint (2 a, 0) {\displaystyle (2a,0)} and radius 2 a {\displaystyle 2a}. The tangent at circle point (2 a + 2 a cos φ, 2 a sin φ) {\displaystyle (2a+2a\cos \varphi,2a\sin \varphi)} has the equation (x − 2 a) ⋅ cos φ + y ⋅ sin φ = 2 a. {\displaystyle (x-2a)\cdot \cos \varphi +y\cdot \sin \varphi =2a\,.} The foot of the perpendicular from point O {\displaystyle O} on the tangent is point (r cos φ, r sin φ) {\displaystyle (r\cos \varphi,r\sin \varphi)} with the still unknown distance r {\displaystyle r} to the origin O {\displaystyle O}. Inserting the point into the equation of the tangent yields (r cos φ − 2 a) cos φ + r sin 2 φ = 2 a → r = 2 a (1 + cos φ) {\displaystyle (r\cos \varphi -2a)\cos \varphi +r\sin ^{2}\varphi =2a\quad \rightarrow \quad r=2a(1+\cos \varphi)} which is the polar equation of a cardioid.

Remark: If point O {\displaystyle O} is not on the perimeter of the circle k {\displaystyle k}, one gets a limaçon of Pascal.