The conjecture that no triangle has rational sides, medians and altitudes I have found a conjecture that there is no triangle whose sides, medians, altitudes and area are all rational. I figure that someone must have already found such a triangle if one existed and yet I saw this conjecture but haven't come across any proof or hints. Is this still an open question? I feel that maybe it can't be proven - is this a question that can't be proven? But how would we show it's logically independent of the standard assumptions in mathematics?
The source where i saw this conjecture is:-https://www.ics.uci.edu/~eppstein/junkyard/open.html
 A: According to this, it was open when Unsolved Problems in Number Theory was published, which was probably in the eighties, although I can't find the publication date for the life of me. Two rational medians, rational sides, and rational area is possible according to this source. Your problem though in fact was still open in 2004 for the third edition of Unsolved Problems in Number Theory by Guy. It is problem D21 if you can get a copy.
Here are the references listed at the end of the section for it.
Roger C. Alperin, A quartic surface of integer hexahedra, Rocky Mountain J.
    Math., 31(2001) 37-43; MR 2002k:11037.
J. H. J. Almering, Heron problems, thesis, Amsterdam, 1950.
A. S. Anema, Pythagorean triangles with equal perimeters, Scripta Math.,
    15(1949) 89.
Raymond A. Beauregard & E. R. Suryanarayan, Arithmetic triangles, Math.
    Mag., 70(1997) 105-115; MR 98d:11033.
Albert H. Beiler, Recreations in the Theory of Numbers - The Queen of Mathematics
    Entertains, Dover, 1964, pp.131-132.
Ralph Heiner Buchholz, On triangles with rational altitudes, angle bisectors
    or medians, PhD thesis, Univ. of Newcastle, Australia, 1989.
Ralph H. Buchholz, Triangles with three rational medians, J. Number Theory,
    97(2002) 113-131; MR 2003h:11034. 

I'm sorry I can't answer your other questions very well, I'm not a logician. However, as I understand it you would have to construct two systems (that are consistent assuming the usual axioms are) in which the usual axioms are satisfied, one in which it is possible to prove this conjecture, and one in which the conjecture is impossible. Anyone with better knowledge please correct me if I'm wrong.
A: A triangle with rational sides and medians is $87,85,68$ (sides), $79,65.5,63.5$ (medians) but irrational area $\sqrt{7207200}$.
A triangle with 3 rational sides, 2 rational medians and a rational area is $73,51,26$ (sides) $48.5,17.5,\sqrt{3796}$ (medians) and area $420$.
And the altitudes will be only rational if both sides and area are rational.
More Info on:

Perfect Rational Triangle
Randall Rathbun
NCR Corporation, Rancho Bernardo

