Binary diagram and two phase region

I know that to find the composition in a two phase region of a binary diagram we have to draw the isotherm and to look at the intersection with phase boundaries. But why? Is there a demonstration?

Consider the physical system: a mixture of two species in two phases is at equilibrium. Gibbs' phase rule tells us that we have two degrees of freedom, which can be chosen from the set $\{P, T, X_B^{liq}, X_B^\text{solid}\}$. By fixing $P$ (first degree of freedom) and letting $T$ (second degree of freedom) vary over a prescribed range, we produce the functions $X_B^{liq} = X_B^{liq}(T)$ and $X_B^\text{solid} = X_B^\text{solid}(T)$, and this is exactly what is represented by a phase diagram.
• I understand this but for $\ X_{b}$ fixed and T fixed (the green dot), why $\ X_{b}^{liquid}$ and $\ X_{b}^{solid}$ gives us the composition in this precise case? I don't know if I'm clear – Hugues Sep 25 '17 at 8:25
• Actually, here you probably have $P$ and $T$ fixed, and $X_B$ just tells you whether or not you're at equilibrium. $X_B$ is not one of the degrees of freedom available because it doesn't tell you anything about phase compositions. For $P$ and $T$ fixed, I have argued that there is only one possible value of $X_B^\text{liq}$ and $X_B^\text{solid}$, and this can be graphically found by looking at the intersection of the curves with the line $T = T^*$, where $T^*$ is your known temperature. – a-cyclohexane-molecule Sep 25 '17 at 14:40
• The last statement is a simple property of functions: if we have a function $f(x)$, then $f(5)$ is the intersection of $f(x)$ with the line $x=5$. – a-cyclohexane-molecule Sep 25 '17 at 14:44