User:Fjackson/Sandbox

Nota Bene. This sandbox page is under construction and may contain inaccuracies.

$\sum_\text{cyclic} \frac {1} {a^2} = \frac{1} {4R^2S^2} (S^2+S_\omega^2) \,$

$\sum_\text{cyclic} \frac {1} {a} = \frac{1} {4r^2RS} (S^2-2r^2S_\omega) \,$ where r is the inradius

$\sum_\text{cyclic} a = \frac{S} {r} \,$

$\sum_\text{cyclic} a^2 = 2 S_\omega \,$

$\sum_\text{cyclic} a^3 = \frac{2S} {r^3}(12Rr^3 + 6S_\omega r^2 - S^2) \,$

$\sum_\text{cyclic} a^4 = 2 (S_\omega^2 - S^2) \,$

$\sum_\text{cyclic} a^6 = 2 (S_\omega^3 - 3S_\omega S^2 + 6R^2 S^2) \,$

$\sum_\text{cyclic} \frac {1} {S_A^2} = \frac{(S^2 - 2S_\omega^2 + 8R^2S_\omega)} {S^2(S_\omega - 4R^2)} \,$

$\sum_\text{cyclic} \frac {1} {S_A} = \frac{1} {S_\omega - 4R^2} \,$

$\sum_\text{cyclic} S_A = S_\omega \,$

$\sum_\text{cyclic} S_A^2 = S_\omega^2- 2S^2 \,$

$\sum_\text{cyclic} S_A^3 = S_\omega^3- 12R^2S^2 \,$

$\sum_\text{cyclic} S_A^4 = S_\omega^4 + 2S^2 + 16R^2 S_\omega S^2 \,$

$\sum_\text{cyclic} \frac {1} {b^2c^2} = \frac {S_\omega} {2S^2R^2} \,$

$\sum_\text{cyclic} \frac {1} {bc} = \frac {1} {2rR} \,$

$\sum_\text{cyclic} bc = \frac {1} {2r^2} (S^2 - 2r^2S_\omega) \,$

$\sum_\text{cyclic} b^2c^2 = S_\omega^2 + S^2 \,$