Show that the central heart shape figure is Cardioid

Question

Can we use Mathematica to show (prove) that the central heart shape figure is Cardioid?

With[{θ = 2.8}, 
Manipulate[
Graphics[Table[{White, Circle[{0, 0}, 0.72], 
Circle[{0, 0}, 1], {Blue, PointSize[0.04], 
Point[{Cos[t + θ], Sin[t + θ]}]}, {Yellow, 
PointSize[0.075], Point[{0, 0}]}, {RGBColor[0.85, 0.87,0.62],
PointSize[0.026],Point[{0.72 Cos[1.62 t + θ], 
0.72 Sin[1.62 t + θ]}]}, Red, 
Line[{{Cos[t + θ], 
Sin[t + θ]}, {0.72 Cos[1.62 t + θ], 
0.72 Sin[1.62 t + θ]}}]}, {t, 0, n, 0.05}], 
Background -> Black], {n, 0, 12.35}]]

Answer 1

Using the usual formula for an envelope:

lin = Rationalize[{{Cos[t + θ], Sin[t + θ]},
                   {0.72 Cos[1.62 t + θ], 0.72 Sin[1.62 t + θ]}}];

{x, InterpolatingPolynomial[lin, x]} /. 
First[Solve[D[InterpolatingPolynomial[lin, x], t] == 0, x]] // FullSimplify
   {(9 (2025 Cos[(19 t)/50 + θ] - 2016 Cos[t + θ] - 475 Cos[(81 t)/50 + θ] +
     900 Cos[(56 t)/25 + θ]))/(-57494 + 58950 Cos[(31 t)/50]),
    (9 (2025 Sin[(19 t)/50 + θ] - 2016 Sin[t + θ] - 475 Sin[(81 t)/50 + θ] +
     900 Sin[(56 t)/25 + θ]))/(-57494 + 58950 Cos[(31 t)/50])}

whose plot with $\theta=\frac{14}{5}$ has only a slight resemblance to a cardioid: