TITLE: Two queries on triangles, the sides of which have rational lengths
QUESTION [7 upvotes]: Let us define a "rational triangle" as one in the Euclidean plane, with lengths of all sides rational.
We are aware that a positive integer is called "congruent" only if it is the area of a right triangle with rational length sides; so not every rational number is the area of some right rational triangle. However,

is every positive rational number the area of some (not necessarily right) rational triangle?
for two given rational numbers $A$ and $P$, among the infinitely many general triangles with area $A$ and perimeter $P$ (there are infinitely many such triangles if $A$ and $P$ are within a suitable range), is there a guarantee that there are any (or infinitely many) rational triangles?

REPLY [6 votes]: If for the equation derived by Chris Wuthrich, we introduce the new variables $p = 2P$, $q = x - P/2$, $r = y-P/2$, we obtain the more symmetric relation $$
pqr(p+q+r) = A^2.
$$
Now I knew I recognized this equation from somewhere, and after a little searching I hit upon some slides by Noam Elkies where your Question 1 is resolved completely. (Elkies further credits Franz Lemmermeyer with bringing this problem under his attention.)
It seems that the credit for this goes to Euler. The surface described by the equation above has a K3 surface with maximal Picard rank 
($\rho=20$) as its smooth model. Euler presumably used the elliptic fibrations and the presence of many rational curves on the surface to produce a $1$-parameter family of solutions (as suggested by Will Sawin).