Sides The heptagonal triangle's sides
a \begin{align} a^2 & =c(c-b), \\[5pt] b^2 & =a(c+a), \\[5pt] c^2 & =b(a+b), \\[5pt] \frac 1 a & =\frac 1 b + \frac 1 c \end{align} (the latter :\frac{a^2}{bc}, \quad -\frac{b^2}{ca}, \quad -\frac{c^2}{ab} satisfy the
cubic equation :t^3+4t^2+3t-1=0. We also have :a^4+b^4+c^4=21R^4. :a^6+b^6+c^6=70R^6. The ratio
r /
R of the
inradius to the circumradius is the positive solution of the cubic equation :8x^3+28x^2+14x-7=0. In addition, :\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=\frac{2}{R^2}. We also have :\frac{1}{a^4}+\frac{1}{b^4}+\frac{1}{c^4}=\frac{2}{R^4}. :\frac{1}{a^6}+\frac{1}{b^6}+\frac{1}{c^6}=\frac{17}{7R^6}. In general for all integer
n, :a^{2n}+b^{2n}+c^{2n}=g(n)(2R)^{2n} where :g(-1) = 8, \quad g(0)=3, \quad g(1)=7 and :g(n)=7g(n-1)-14g(n-2)+7g(n-3). We also have :2b^2-a^2=\sqrt{7}bR, \quad 2c^2-b^2=\sqrt{7}cR, \quad 2a^2-c^2=-\sqrt{7}aR. We also have :a^{3}c + b^{3}a - c^{3}b = -7R^{4}, :a^{4}c - b^{4}a + c^{4}b = 7\sqrt{7}R^{5}, :a^{11}c^{3}+b^{11}a^{3} - c^{11}b^{3} = -7^{3}17R^{14}. The
exradius ra corresponding to side
a equals the radius of the
nine-point circle of the heptagonal triangle. ==Orthic triangle==