Most standard textbooks on line transmission cover
this. It is derived from the T equivalent circuit
of the transmission line, having total series impedance
of Z1 and shunt impedance z2, correctly terminated
in Z0.

Looking from the left end as the input
with the right end short circuited, the input impedance
is

Zsc = Z1/2 + Z1*Z2/(Z1 + 2*Z2) .............1

Open circuited, the input impedance at left
is

Zoc = Z1/2 + Z2 ......................................2

The Characteristic Impedance is given by

Z0 = SQRT( Z1*Z2 + Z1^2/4) ..................3

which needs a separate proof.

If equations
1 and 2 are multiplied together you get

Zoc
* Zsc = Z1^2/4 + Z1 * Z2 = Z0^2

Thus Z0 =
SQRT (Zsc * Zoc)

My textbook is an ancient
Everitt "Communication Engineering" written with
a quill pen on parchment.