At altitudes from 12-20 km (7-12.5 mi) above the midlatitudes the temperature remains constant with altitude (minus 56.5°C in the ISA). This part is usually considered to be in the stratosphere, but is there a special reason for that, or wouldn't it matter if the tropopause was set at the upper boundary of the isothermal layer rather than the lower one? There's another isothermal layer, the stratopause at 48-52 km (29.8-32.3 mi), here we set the stratopause simply at its center, to 50 km. Why isn't the tropopause set to the center of its isothermal layer as well, but on its lower boundary?
This is simply because of the way that the troposphere is defined and the fact that isothermal layers are of constant temperature. The troposphere has one defining characteristic; the air temperature drops with increased altitude. The isothermal layer is where that stops happening and temperatures remain stable with increased altitude, so the tropopause is at the base of it.
The stratopause is different because it is defined as the point of maximum temperature of both the stratosphere and the mesosphere. It is traditionally represnted as a line placed in the middle of the upper stratospheric isothermal layer as the halfway mark between the two layers but in reality the entire isothermal layer is, by definition, the stratopause.