While I can't be sure, since you have provided no context or links to the original source, I would guess that this is part of a discussion on how to quantify 3 terms: input high, input low, and mid-voltage.
What the author seems to have done is to take the position that, for a logic circuit, there will be input/output high and low regimes, where changing the input has little or no effect on the output, and a middle area where the output is sensitive to changes in input. Apparently the author has chosen to define the input high and low thresholds as those at which the change in input is equal to change in output - where the slope of input vs output is -1. The mid-point, where the input equals the output, is obviously where the curve intersects the line Vin = Vout; in otherwords where it intersects a line with slope = 1 and which intersects the origin.
This is a generalized approach to analyzing response curves, and has the advantage of dealing in a consistent way with curves which are not ideal. An ideal inverter, for instance, has an input/output curve with flat in the high and low regions and a vertical transition region. For such a devices, Vil, Vm, and Vih are the same. But real devices don't have infinite gain, and your figure seems to be one way to deal with this.
Note that this is not the only way to deal with the problem. You might, for instance, define the transition points as those which produce outputs of 10% and 90% of the output range, and Vm as the 50% point. Or any other set of values you like.