Cardioid
Proof of C1

The points P: p (φ), Q: p (φ + π) {\displaystyle P:p(\varphi),\;Q:p(\varphi +\pi)} are on a chord through the cusp (=origin). Hence | P Q | = r (φ) + r (φ + π) = 2 a (1 − cos φ) + 2 a (1 − cos (φ + π)) = ⋯ = 4 a. {\displaystyle {\begin{aligned}|PQ|&=r(\varphi)+r(\varphi +\pi)\\&=2a(1-\cos \varphi)+2a(1-\cos(\varphi +\pi))=\cdots =4a\end{aligned}}.}
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