I have this circuit where Rt represents an NTC sensor and I have this curve:

T is the temperature.
In the figure I already wrote the expression V(T) of the linearized response (green line) .. but I do not understand how it was calculated.
I'm not interested in numerical calculations and I didn't even write down the data, I'm interested in how that curve was 'Linearised' around (I assume) the Vs/2 point and Tx

 

The equation of the lines is derived from a point on the curve with coordinates (Tx, Vs/2) and the slope of the curve at that point. If (x0, y0) is the point and m is the slope then the equation of the line is:

\( y-y_0\;=\;m(x-x_0) \)

Simple algebra will allow you to rearrange that in the more familiar form:

\( y\;=\;mx+b \)