I haven't watched the video so I don't know exactly how he's defined the thresholds; but...
The main problem with positive feedback is, you don't have one threshold, it splits into two, the rising and falling thresholds.
The analysis is simple. When the comparator is in the output-high state, evaluate the node voltages and all that. You'll see the voltage gain (meaning, from Vin to +in) is defined by resistor ratios, and the threshold by Vout, Vref and some resistor ratios. Repeat the process for the output-low state.
Once you've done that, subtract the two thresholds, and you have the hysteresis band Vhyst.
Now, if you prefer to think in terms of a single threshold, you can average them together. The meaning of a single threshold with hysteresis is, at Vth, you must either add or subtract slightly more than Vhyst/2 to cause the comparator to change state. It's a symmetry point, because the amount added or subtracted, that causes it to change state, is equal.
Just from your description anyway, I don't understand which definition he's using, because you only mention a single threshold, no hysteresis band, and none of the relations depend on Vout. As I say above, it is possible to define a meaningful "threshold" (that is, a single one) for such a circuit, but it's not clear if this is what is meant; and in any case, it is insufficient information, because two thresholds means two thresholds, period, no matter what: a complete definition must always be either: the two thresholds as such, or the average with hysteresis band.
Note that, since the threshold depends on the output voltage, it further matters what type of comparator is used, and what circuit it is used within. Comparators are often open-collector output types, which require a pull-up. If the pull-up is a resistor (and the load doesn't affect the voltage at all), then this resistor is in series with the feedback resistor when the output is high, but not when it is low. If the output voltage is constrained (a typical application would be: the OC comparator drives a bipolar transistor base, which means it's clamped to +0.7V when "high"), then that must be used in the analysis. If the output is active driven (typical of some general-purpose comparators, and many high-speed comparators), you use V_OH and V_OL in the analysis.
Tim