Conway triangle notation

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

${\displaystyle S=bc\sin A=ac\sin B=ab\sin C\,}$

where S = 2 × area of reference triangle and

${\displaystyle S_{\varphi }=S\cot \varphi .\,}$

in particular

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

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

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

${\displaystyle S_{\omega }=S\cot \omega ={\frac {a^{2}+b^{2}+c^{2}}{2}}\,}$     where ${\displaystyle \omega \,}$ is the Brocard angle. The law of cosines is used: ${\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A}$.

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

${\displaystyle S_{2\varphi }={\frac {S_{\varphi }^{2}-S^{2}}{2S_{\varphi }}}\quad \quad S_{\frac {\varphi }{2}}=S_{\varphi }+{\sqrt {S_{\varphi }^{2}+S^{2}}}\,}$   for values of  ${\displaystyle \varphi }$  where  ${\displaystyle 0<\varphi <\pi \,}$

${\displaystyle S_{\vartheta +\varphi }={\frac {S_{\vartheta }S_{\varphi }-S^{2}}{S_{\vartheta }+S_{\varphi }}}\quad \quad S_{\vartheta -\varphi }={\frac {S_{\vartheta }S_{\varphi }+S^{2}}{S_{\varphi }-S_{\vartheta }}}\,.}$

Furthermore the convention uses a shorthand notation for ${\displaystyle S_{\vartheta }S_{\varphi }=S_{\vartheta \varphi }\,}$ and ${\displaystyle S_{\vartheta }S_{\varphi }S_{\psi }=S_{\vartheta \varphi \psi }\,.}$

Hence:

${\displaystyle \sin A={\frac {S}{bc}}={\frac {S}{\sqrt {S_{A}^{2}+S^{2}}}}\quad \quad \cos A={\frac {S_{A}}{bc}}={\frac {S_{A}}{\sqrt {S_{A}^{2}+S^{2}}}}\quad \quad \tan A={\frac {S}{S_{A}}}\,}$

${\displaystyle a^{2}=S_{B}+S_{C}\quad \quad b^{2}=S_{A}+S_{C}\quad \quad c^{2}=S_{A}+S_{B}\,.}$

Some important identities:

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

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

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

${\displaystyle S_{ABC}=S_{A}S_{B}S_{C}=S^{2}(S_{\omega }-4R^{2})\quad \quad S_{\omega }=s^{2}-r^{2}-4rR\,}$

where R is the circumradius and abc = 2SR and where r is the incenter,  ${\displaystyle s={\frac {a+b+c}{2}}\,}$   and  ${\displaystyle a+b+c={\frac {S}{r}}\,.}$

Some useful trigonometric conversions:

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

${\displaystyle \sum _{\text{cyclic}}\sin A={\frac {S}{2Rr}}={\frac {s}{R}}\quad \quad \sum _{\text{cyclic}}\cos A={\frac {r+R}{R}}\quad \quad \sum _{\text{cyclic}}\tan A={\frac {S}{S_{\omega }-4R^{2}}}=\tan A\tan B\tan C\,.}$

Some useful formulas:

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

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

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_q = aq_a + bq_b + cq_c. Then D is given by the formula:

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

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 = S_BS_C/a

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

Hence:

${\displaystyle {\begin{aligned}D^{2}&{}=\sum _{\text{cyclic}}a^{2}S_{A}\left({\frac {aS_{A}}{2S^{2}}}-{\frac {S_{B}S_{C}}{aS^{2}}}\right)^{2}\\&{}={\frac {1}{4S^{4}}}\sum _{\text{cyclic}}a^{4}S_{A}^{3}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}+{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}S_{B}S_{C}\\&{}={\frac {1}{4S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}^{2}(S^{2}-S_{B}S_{C})-2(S_{\omega }-4R^{2})+(S_{\omega }-4R^{2})\\&{}={\frac {1}{4S^{2}}}\sum _{\text{cyclic}}a^{2}S_{A}^{2}-{\frac {S_{A}S_{B}S_{C}}{S^{4}}}\sum _{\text{cyclic}}a^{2}S_{A}-(S_{\omega }-4R^{2})\\&{}={\frac {1}{4S^{2}}}\sum _{\text{cyclic}}a^{2}(b^{2}c^{2}-S^{2})-{\frac {1}{2}}(S_{\omega }-4R^{2})-(S_{\omega }-4R^{2})\\&{}={\frac {3a^{2}b^{2}c^{2}}{4S^{2}}}-{\frac {1}{4}}\sum _{\text{cyclic}}a^{2}-{\frac {3}{2}}(S_{\omega }-4R^{2})\\&{}=3R^{2}-{\frac {1}{2}}S_{\omega }-{\frac {3}{2}}S_{\omega }+6R^{2}\\&{}=9R^{2}-2S_{\omega }.\end{aligned}}}$

This gives:

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

References

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