f[x_] := x^2
g[x_] := (x - 2)^2 + 3

sol = Solve[(f[x1] - g[x2])/(x1 - x2) == f'[x1] == g'[x2], {x1, x2}, Reals]

(* ==> {{x1 -> 3/4, x2 -> 11/4}} *)

eqns = FlattenAt[{f[x], g[x], f'[x1] x + g[x2] - f'[x1] x2 /. sol}, 3];
Plot[eqns, {x, -2, 4}, Frame -> True, Axes -> None]

↑ に　類比な　次の　放物線　c1,c2

も　(5)   ↑　の　発想で　速やかに　解いて下さい；