Conway triangle notation

In geometry, the Conway triangle notation allows trigonometric functions of a triangle to be managed algebraically. Given a reference triangle whose sides are a, b and c and whose internal angles are A, B, and C then the Conway triangle notation is simply represented as follows:

$S=bc\sin A=ac\sin B=ab\sin C\,$    where    $S\,$    is 2 × Area of reference triangle and

$S_{\phi }=S\cot \phi .\,$

in particular

$S_{A}=S\cot A=bc\cos A={\frac {b^{2}+c^{2}-a^{2}}{2}}\,$

$S_{B}=S\cot B=ac\cos B={\frac {a^{2}+c^{2}-b^{2}}{2}}\,$

$S_{C}=S\cot C=ab\cos C={\frac {a^{2}+b^{2}-c^{2}}{2}}\,$

$S_{\omega }=S\cot \omega ={\frac {a^{2}+b^{2}+c^{2}}{2}}\,$      where $\omega \,$ is the Brocard angle.

$S_{\frac {\pi }{3}}=S\cot {\frac {\pi }{3}}=S{\frac {\sqrt {3}}{3}}\,$

Hence:

$\sin A={\frac {S}{bc}}\quad \quad \cos A={\frac {S_{A}}{bc}}\quad \quad \tan A={\frac {S}{S_{A}}}\,$

Some important identities:

$\sum _{\text{cyclic}}S_{A}=S_{A}+S_{B}+S_{C}=S_{\omega }\,$

$S^{2}=b^{2}c^{2}-S_{A}^{2}=a^{2}c^{2}-S_{B}^{2}=a^{2}b^{2}-S_{C}^{2}\,$

$S_{B}S_{C}=S^{2}-a^{2}S_{A}\quad \quad S_{A}S_{C}=S^{2}-b^{2}S_{B}\quad \quad S_{A}S_{C}=S^{2}-c^{2}S_{C}\,$

$S_{A}S_{B}S_{C}=S^{2}(S_{\omega }-4R^{2})\,$

where R is the circumradius and $abc=2SR.\,$

Some useful trigonometric conversions:

$\sin A\sin B\sin C={\frac {S}{4R^{2}}}\quad \quad \cos A\cos B\cos C={\frac {S_{\omega }-4R^{2}}{4R^{2}}}$

$\sum _{\text{cyclic}}\sin A={\frac {S}{2Rr}}\quad \quad \sum _{\text{cyclic}}\cos A={\frac {S^{2}-4S_{\omega }r^{2}-12Rr^{3}}{4Rr^{3}}}\,$    where $r\,$ is the incenter and    $a+b+c={\frac {S}{r}}\,$

Some useful formulas:

$\sum _{\text{cyclic}}a^{2}S_{A}=a^{2}S_{A}+b^{2}S_{B}+c^{2}S_{C}=2S^{2}\quad \quad \sum _{\text{cyclic}}a^{4}=2(S_{\omega }^{2}-S^{2})\,$

$\sum _{\text{cyclic}}S_{A}^{2}=S_{\omega }^{2}-2S^{2}\quad \quad \sum _{\text{cyclic}}S_{B}S_{C}=S^{2}\quad \quad \sum _{\text{cyclic}}b^{2}c^{2}=S_{\omega }^{2}+S^{2}\,$

Some examples using Conway triangle notation:

Let D be the distance between two points P and Q whose trilinear coordinates are p_a : p_b : p_c and q_a : q_b : q_c. Let K_p = ap_a + bp_b + cp_c and let K_p = aq_a + bq_b + cq_c. Then D is given by the formula:

$D^{2}=\sum _{\text{cyclic}}a^{2}S_{A}\left({\frac {p_{a}}{K_{p}}}-{\frac {q_{a}}{K_{q}}}\right)^{2}\,$

Using this formula it is possible to determine OH, the distance between the circumcenter and the orthocenter as follows:

For the circumcenter $p_{a}=aS_{A}\,$ and for the orthocenter $q_{a}={\frac {S_{B}S_{C}}{a}}\,$

$K_{p}=\sum _{\text{cyclic}}a^{2}S_{A}=2S^{2}\quad \quad K_{q}=\sum _{\text{cyclic}}S_{B}S_{C}=S^{2}\,$

Hence:

$D^{2}=\sum _{\text{cyclic}}a^{2}S_{A}\left({\frac {aS_{A}}{2S^{2}}}-{\frac {S_{B}S_{C}}{aS^{2}}}\right)^{2}\,$

$={\frac {1}{4S^{4}}}\sum _{\text{cyclic}}a^{4}S_{A}^{3}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}+{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}S_{B}S_{C}\,$

$={\frac {1}{4S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}^{2}(S^{2}-S_{B}S_{C})-2(S_{\omega }-4R^{2})+(S_{\omega }-4R^{2})\,$

$={\frac {1}{4S^{2}}}\sum _{\text{cyclic}}a^{2}S_{A}^{2}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}-(S_{\omega }-4R^{2})\,$

$={\frac {1}{4S^{2}}}\sum _{\text{cyclic}}a^{2}(b^{2}c^{2}-S^{2})-{\frac {1}{2}}(S_{\omega }-4R^{2})-(S_{\omega }-4R^{2})\,$

$={\frac {3a^{2}b^{2}c^{2}}{4S^{2}}}-{\frac {1}{4}}\sum _{\text{cyclic}}a^{2}-{\frac {3}{2}}(S_{\omega }-4R^{2})\,$

$=3R^{2}-{\frac {1}{2}}S_{\omega }-{\frac {3}{2}}S_{\omega }+6R^{2}\,$

$=9R^{2}-2S_{\omega }\,$

Which gives:

$OH={\sqrt {9R^{2}-2S_{\omega }\,}}$

References

• Weisstein, Eric W. "Conway Triangle Notation". MathWorld.