Properties of azeotropes
I will first enumerate the four points that need to be addressed.


The vapor and liquid phases have the same composition in an azeotrope.
The azeotrope boils at a constant temperature.
The composition of the azeotrope remains fixed while boiling.
The azeotropic mixture cannot be separated by fractional distillation.


I will assume the definition of an azeotrope to be a solution whose composition is such that it exists at a local extremum of a Pxy or Txy plot. I prefer to instead take 1. as a definition, in which case my original definition follows as a property of an azeotrope, but both approaches are equivalent.

"Proof" of 1., by  contradiction. Overlay the corresponding vapor pressure curve as a function of $y_A$ to make your plot a Pxy plot for species $A$. This new vapor pressure curve must touch the original curve at the azeotropic composition $x^*=y^*=z$, for otherwise a tie line would connect two vapor phases at some $P$, which is physically unreasonable.

Proof of 2. Refer to the Pxy plot from 1. The dew point and bubble point coincide at the azeotrope, so a phase transition happens only at one value of $P$ and hence only at one value of $T$.

Proof of 3. Refer to the Pxy plot from 1. The dew point and bubble point coincide at the azeotrope, and so must the compositions of each phase.

Proof of 4. Fractional distillation is based on the composition of the vapor and liquid phases being different. This is precluded by 3. in an azeotrope.