Writing:

`{c1, c2, c3, c4, c5} = N[{Tan[E], Sin[E], Tanh[E], E, Sinh[E]}]; a = (c1 + c3) / 2; b = Sqrt[c5^2 - (c2 - c4)^2] / 2; c = 0; d = (c2 + c4) / 2; e = (c1 - c3) (c2 - c4) / (4 b); f = c5 Sqrt[c5^2 - (c1 - c3)^2 - (c2 - c4)^2] / (4 b); x = a + b Cos[t] + c Sin[t]; y = d + e Cos[t] + f Sin[t]; xmin = Minimize[{x, 0 <= t <= 2π}, t][[1]]; xmax = Maximize[{x, 0 <= t <= 2π}, t][[1]]; FullSimplify[(xmin + xmax) / 2 == a] ymin = Minimize[{y, 0 <= t <= 2π}, t][[1]]; ymax = Maximize[{y, 0 <= t <= 2π}, t][[1]]; FullSimplify[(ymin + ymax) / 2 == d] `

we get:

True

True

which is what is desired. On the other hand, by making a simple change:

`{c1, c2, c3, c4, c5} = {Tan[E], Sin[E], Tanh[E], E, Sinh[E]}; `

we get:

True

…

that is, in the second case there is no answer. So by defining the constants in this other way:

`SetAttributes[c1, Constant] NumericQ[c1] = True; N[c1, prec___] := N[Tan[E], prec] SetAttributes[c2, Constant] NumericQ[c2] = True; N[c2, prec___] := N[Sin[E], prec] SetAttributes[c3, Constant] NumericQ[c3] = True; N[c3, prec___] := N[Tanh[E], prec] SetAttributes[c4, Constant] NumericQ[c4] = True; N[c4, prec___] := N[E, prec] SetAttributes[c5, Constant] NumericQ[c5] = True; N[c5, prec___] := N[Sinh[E], prec] `

we get:

True

Minimize::infeas: There are no values of {t} for which the constraints 0<=t<=2π are satisfied and the objective function […] is real-valued.

Maximize::infeas: There are no values of {t} for which the constraints 0<=t<=2π are satisfied and the objective function […] is real-valued.

Infinity::indet: Indeterminate expression -∞+∞ encountered.

Indeterminate == (c2 + c4) / 2

where, apparently, another problem arises. How to solve it all?