Finding the intersection of two parametrised functions

I beg to differ :)

Using Bob Hanlon's set-up, an exact solution, numerically equal to Feyre's:

Block[{a, b, c, cy, ey, cx, ex, ix, iy},
 a = 1;
 ex[t_] = a Cos[t] + a t Sin[t];
 ey[t_] = a Sin[t] - a t Cos[t];
 b = 3/2;
 cx[t_] = b Cos[t];
 cy[t_] = b Sin[t];
 c = 1;
 ix[t_] = c Cos[t];
 iy[t_] = c Sin[t];

 sol = Solve[
  {ex[t1] == cx[t2], ey[t1] == cy[t2], 0 < t1 < Pi/2, 0 < t2 < Pi/2},
  {t1, t2}, Method -> Reduce]
 ];
sol
N[sol]

Remarks: To get the intersection of two curves, you need two parameters as the other answers show. The best way, imo, is to make the expressions explicit functions of the parameters (but one could use ReplaceAll: ex /. t -> t1, etc.). Solving transcendental equations exactly is often not possible, but when trying, it helps to limit the domains of the variables and try Reduce[].

Redefine with different variables:

cx := b Cos[u]
cy := b Sin[u]

Minimize, like @J.M. said, this won't give exact results.

NMinimize[Sqrt[(cy - ey)^2 + (cx - ex)^2], {u, t}, Reals]

{6.59602*10^-10, {u -> 0.276965, t -> 1.11803}}

{ex, ey} /. {u -> 0.27696531800957025`, t -> 1.1180339888944713`}
{cx, cy} /. {u -> 0.27696531800957025`, t -> 1.1180339888944713`}

{1.44283, 0.410157}

{1.44283, 0.410157}

Returning to single parameter:

Show[ParametricPlot[{{ex, ey}, {cx, cy}, {ix, iy}}, {t, 0, π/2}, 
  PlotLabel -> "Evolute", PlotLegends -> "Expressions"], 
 Graphics[{Red, PointSize[0.02], 
   Point[{1.4428344948227565`, 0.41015682473030135`}]}]]

a = 1;
ex[t_] = a Cos[t] + a t Sin[t];
ey[t_] = a Sin[t] - a t Cos[t];
b = 3/2;
cx[t_] = b Cos[t];
cy[t_] = b Sin[t];
c = 1;
ix[t_] = c Cos[t];
iy[t_] = c Sin[t];

pt = {ex[t1], ey[t1]} /.
  FindRoot[
   {ex[t1] == cx[t2], ey[t1] == cy[t2]},
   {{t1, π/4}, {t2, π/4}}]

(*  {1.44283, 0.410157}  *)

ParametricPlot[
 {{ex[t], ey[t]}, {cx[t], cy[t]}, {ix[t], iy[t]}},
 {t, 0, π/2},
 PlotLabel -> "Evolute",
 PlotLegends -> "Expressions",
 Epilog -> {Red, AbsolutePointSize[4],
   Tooltip[Point[pt], pt]}]