Smith chart of λ/4 Transmission Line
Why traverse across the upper half circle ?


When translating an impedance towards the generator we move clockwise around the chart.

When translating an impedance towards the load, we move counter-clockwise around the chart.


  Why "ZT-line is the geometric mean of Zout and Zin." ?


This is a design decision that was made to get the desired result. If you choose a different \$Z_T\$, you won't get the equivalent load (\$Z_T\$ line plus actual load) to match your input impedance.

Notice how they overlaid the Smith chart scaled for \$Z_T\$ on top of the \$Z_0\$ Smith chart. You need to be able to rotate the load around from the 0-ohm point to the \$\infty\$-ohm point on the \$Z_T\$ Smith chart and have that end up being the correct place on the \$Z_0\$ Smith chart. That only happens if you choose \$Z_T=\sqrt{Z_{out} Z_{in}}\$