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:

:$S = bc \sin A = ac \sin B = ab \sin C \,$

where ''S'' = 2 × area of reference triangle and

:$S_\varphi = S \cot \varphi . \,$

in particular

:$S_A = S \cot A = bc \cos A= \frac \,$

:$S_B = S \cot B = ac \cos B= \frac \,$

:$S_C = S \cot C = ab \cos C= \frac \,$

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

:$S_ = S \cot = S \frac \,$

:$S_ = \frac \quad\quad S_ = S_\varphi + \sqrt \,$    for values of   $\varphi$  where   $0 < \varphi < \pi \,$

:$S_ = \frac \quad\quad S_ = \frac \, .$

Furthermore the convention uses a shorthand notation for $S_S_=S_ \,$ and $S_S_S_=S_ \, .$

Hence:

:$\sin A = \frac = \frac \quad\quad \cos A = \frac = \frac \quad\quad \tan A = \frac \,$

:$a^2 = S_B + S_C \quad\quad b^2 = S_A + S_C \quad\quad c^2 = S_A + S_B \, .$

Some important identities:

:$\sum_\text S_A = S_A+S_B+S_C = S_\omega \,$

:$S^2 = b^2c^2 - S_A^2 = a^2c^2 - S_B^2 = a^2b^2 - S_C^2 \,$

:$S_ = S_BS_C = S^2 - a^2S_A \quad\quad S_ = S_AS_C = S^2 - b^2S_B \quad\quad S_ = S_AS_B = S^2 - c^2S_C \,$

:$S_ = S_AS_BS_C = S^2(S_\omega-4R^2)\quad\quad S_\omega=s^2-r^2-4rR \,$

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

Some useful trigonometric conversions:

:$\sin A \sin B \sin C = \frac \quad\quad \cos A \cos B \cos C = \frac$
:$\sum_\text \sin A = \frac = \frac \quad\quad \sum_\text \cos A = \frac \quad\quad \sum_\text \tan A = \frac =\tan A \tan B \tan C \, .$

Some useful formulas:

:$\sum_\text a^2S_A = a^2S_A + b^2S_B + c^2 S_C = 2S^2 \quad\quad \sum_\text a^4 = 2(S_\omega^2-S^2) \,$

:$\sum_\text S_A^2 = S_\omega^2 - 2S^2 \quad\quad \sum_\text S_ = \sum_\text S_BS_C = S^2 \quad\quad \sum_\text b^2c^2 = S_\omega^2 + S^2 \, .$

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:

:$D^2= \sum_\text a^2S_A\left(\frac - \frac \right)^2 \, .$

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''_''B''''S''_''C''/''a''
:$K_p= \sum_\text a^2S_A = 2S^2 \quad\quad K_q= \sum_\text S_BS_C = S^2 \, .$

Hence:

:$\begin D^2 & = \sum_\text a^2S_A\left(\frac - \frac \right)^2 \\ & = \frac \sum_\text a^4S_A^3 - \frac \sum_\text a^2S_A + \frac \sum_\text S_BS_C \\ & = \frac \sum_\text a^2S_A^2(S^2-S_BS_C) - 2(S_\omega-4R^2) + (S_\omega-4R^2) \\ & = \frac \sum_\text a^2S_A^2 - \frac \sum_\text a^2S_A - (S_\omega-4R^2) \\ & = \frac \sum_\text a^2(b^2c^2-S^2) - \frac (S_\omega-4R^2) -(S_\omega-4R^2) \\ & = \frac - \frac \sum_\text a^2 - \frac (S_\omega-4R^2) \\ & = 3R^2- \frac S_\omega - \frac S_\omega + 6R^2 \\ & = 9R^2- 2S_\omega. \end$

This gives:

:$OH = \sqrt.$

References

* {{mathworld, urlname=ConwayTriangleNotation, title=Conway Triangle Notation

Triangle geometry
Trigonometry
John Horton Conway