Re Fred Lunnon's Heronian triangle problem. Lunnon says the Heronian triangles = the triangles with all 3 sides a,b,c, and the area, being integers. Wikipedia http://en.wikipedia.org/wiki/Heronian_triangle says: Theorem: Given a Heronian triangle, one can split it into two right triangles, whose sidelengths form Pythagorean triples with rational entries. This proves any Heronian triangle can be made to have rational vertex coordinates. Therefore after scale-up by LCM(denominators of coordinates), it can be made to have all coordinates integer. Lunnon: Bucholtz' parameterisation should read a = (n^2 + k^2)m, b = (m^2 + k^2)n, c = (m + n)(m n - k^2), semiperim=s = (m + n)m n, area=d = k m n(m + n)(m n - k^2); constraints GCD(m,n,k) = 1; m > n > 0 & 0 < k <= sqrt( m n^2/(m + 2 n) ) enforce one (nonprimitive) triangle in each Heronian similarity class. The inverse mapping (modulo similarity) is given by k = d, n = s(s-b), m = s(s-a). The wikipedia Pythag proof shows the altitude of the triangle is given by altitude = 2*area / base and the two legs of the right triangles which both lie on the base are ( hypot1^2 - hypot2^2 + base^2 ) / (2*base) and ( hypot2^2 - hypot1^2 + base^2 ) / (2*base). Taking base=c=leg1+leg2 and hypot1=a and hypot2=b we find altitude = 2*k*m*n leg1 = m*(n^2-k^2) leg2 = n*(m^2-k^2) all of which are integers, proving Lunnon's "integer poseability" conjecture. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)