I have been using OriginLab to plot ternary phase diagrams but I want to replace it with an open source package that may have more decent capabilities in that sense.
Last time I checked LabPlot, QtiPlot and its fork SciDAVIS had no option for ternary diagrams.
You can try out Veusz: http://home.gna.org/veusz/examples/#ternary, but it does not look trivial.
I am not aware of any other open source, multi-platform program for ternary diagrams (though it is possible to do it with gnuplot with coordinates transformations and some lines for the axes).
As free but closed source, you could give a try to ProSim Ternary Diagram. It probably works with Wine for Linux users. Not many options, but does the job.
You may download the academic version of ThermoCalc (Validity 365 days and demo databases with upto 3 elements) from https://www.thermocalc.com/academia/students/educational-package-for-students/. Alternatively you can download the ternary thermodynamic database file (.tdb) of interest from NIMS/NIST repository and run it using ThermoCalc academic version. Alternatively you may use PyCalphad or OpenCalphad.