Conway triangle notation

In geometry, the Conway triangle notation simplifies and clarifies the algebraic expression of various trigonometric relationships in a triangle. Using the symbol $S$ for twice the triangle's area, the symbol $S φ$ is defined to mean $S$ times the cotangent of any arbitrary angle $φ$ .

The notation is named after English mathematician John Horton Conway, who promoted its use, but essentially the same notation (using $p$ instead of $S$ ) can be found in an 1894 paper by Spanish mathematician Juan Jacobo Durán Loriga [gl].

Definition

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: $S = b c \sin A = a c \sin B = a b \sin C,$ where $\begin{matrix} S & = 2 \times reference triangle area, \\ S_{φ} & = S \cot φ . \end{matrix}$

Basic formulas

In particular: $\begin{matrix} S_{A} & = S \cot A = b c \cos A = \frac{b^{2} + c^{2} - a^{2}}{2}, \\ S_{B} & = S \cot B = a c \cos B = \frac{a^{2} + c^{2} - b^{2}}{2}, \\ S_{C} & = S \cot C = a b \cos C = \frac{a^{2} + b^{2} - c^{2}}{2}, \\ S_{ω} & = S \cot ω = \frac{a^{2} + b^{2} + c^{2}}{2}, \end{matrix}$

where $ω$ is the Brocard angle. The law of cosines is used: $a^{2} = b^{2} + c^{2} - 2 b c \cos A .$

Third-, double-, and half-angle identities: $\begin{matrix} S_{\frac{π}{3}} & = S \cot \frac{π}{3} = S \frac{\sqrt{3}}{3} \\ S_{2 φ} & = \frac{S_{φ}^{2} - S^{2}}{2 S_{φ}} \\ S_{\frac{φ}{2}} & = S_{φ} + \sqrt{S_{φ}^{2} + S^{2}} \end{matrix}$

for values of $φ$ where $0 < φ < π$ ,

$\begin{matrix} S_{ϑ + φ} & = \frac{S_{ϑ} S_{φ} - S^{2}}{S_{ϑ} + S_{φ}}, \\ S_{ϑ - φ} & = \frac{S_{ϑ} S_{φ} + S^{2}}{S_{φ} - S_{ϑ}} . \end{matrix}$

Furthermore the convention uses a shorthand notation for $S_{ϑ} S_{φ} = S_{ϑ φ},$ and $S_{ϑ} S_{φ} S_{ψ} = S_{ϑ φ ψ} .$

Trigonometric relationships

$\begin{matrix} \sin A & = \frac{S}{b c} = \frac{S}{\sqrt{S_{A}^{2} + S^{2}}} \\ \cos A & = \frac{S_{A}}{b c} = \frac{S_{A}}{\sqrt{S_{A}^{2} + S^{2}}} \\ \tan A & = \frac{S}{S_{A}} \\ a^{2} & = S_{B} + S_{C} \\ b^{2} & = S_{A} + S_{C} \\ c^{2} & = S_{A} + S_{B} \end{matrix}$

Important identities

$\begin{matrix} \sum_{cyclic} S_{A} & = S_{A} + S_{B} + S_{C} = S_{ω} \\ S^{2} & = b^{2} c^{2} - S_{A}^{2} = a^{2} c^{2} - S_{B}^{2} = a^{2} b^{2} - S_{C}^{2} \\ S_{B C} & = S_{B} S_{C} = S^{2} - a^{2} S_{A} \\ S_{A C} & = S_{A} S_{C} = S^{2} - b^{2} S_{B} \\ S_{A B} & = S_{A} S_{B} = S^{2} - c^{2} S_{C} \\ S_{A B C} & = S_{A} S_{B} S_{C} = S^{2} (S_{ω} - 4 R^{2}) \\ S_{ω} & = s^{2} - r^{2} - 4 r R \end{matrix}$

where

R

is the circumradius

r

is the incenter

a b c = 2 S R

s = \frac{a + b + c}{2}

\frac{S}{r} = a + b + c

Trigonometric conversions

$\begin{matrix} \sin A \sin B \sin C & = \frac{S}{4 R^{2}} \\ \cos A \cos B \cos C & = \frac{S_{ω} - 4 R^{2}}{4 R^{2}} \end{matrix}$

$\begin{matrix} \sum_{cyclic} \sin A & = \frac{S}{2 R r} = \frac{s}{R} \\ \sum_{cyclic} \cos A & = \frac{r + R}{R} \\ \sum_{cyclic} \tan A & = \frac{S}{S_{ω} - 4 R^{2}} = \tan A \tan B \tan C \end{matrix}$

Useful formulas

$\begin{matrix} \sum_{cyclic} a^{2} S_{A} & = & a^{2} S_{A} + b^{2} S_{B} + c^{2} S_{C} = 2 S^{2} \\ \sum_{cyclic} a^{4} & = & 2 (S_{ω}^{2} - S^{2}) \\ \sum_{cyclic} S_{A}^{2} & = & S_{ω}^{2} - 2 S^{2} \\ \sum_{cyclic} S_{B C} & = & \sum_{cyclic} S_{B} S_{C} = S^{2} \\ \sum_{cyclic} b^{2} c^{2} & = & S_{ω}^{2} + S^{2} \end{matrix}$

Applications

Let $D$ be the distance between two points $P$ and $Q$ whose trilinear coordinates are $\begin{matrix} P = p_{a} : p_{b} : p_{c}, \\ Q = q_{a} : q_{b} : q_{c} . \end{matrix}$ Let $\begin{matrix} K_{p} & = a p_{a} + b p_{b} + c p_{c}, \\ K_{q} & = a q_{a} + b q_{b} + c q_{c} . \end{matrix}$ Then $D$ is given by the formula: $D^{2} = \sum_{cyclic} a^{2} S_{A} {(\frac{p_{a}}{K_{p}} - \frac{q_{a}}{K_{q}})}^{2}$

Distance between circumcenter and orthocenter

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

For the circumcenter $p_{a} = a S_{A},$

and for the orthocenter $q_{a} = \frac{S_{B} S_{C}}{a},$

$\begin{matrix} K_{p} & = \sum_{cyclic} a^{2} S_{A} = 2 S^{2}, \\ K_{q} & = \sum_{cyclic} S_{B} S_{C} = S^{2} . \end{matrix}$

Hence:

$\begin{matrix} D^{2} & = \sum_{cyclic} a^{2} S_{A} {(\frac{a S_{A}}{2 S^{2}} - \frac{S_{B} S_{C}}{a S^{2}})}^{2} \\ = \frac{1}{4 S^{4}} \sum_{cyclic} a^{4} S_{A}^{3} - \frac{S_{A} S_{B} S_{C}}{S^{4}} \sum_{cyclic} a^{2} S_{A} + \frac{S_{A} S_{B} S_{C}}{S^{4}} \sum_{cyclic} S_{B} S_{C} \\ = \frac{1}{4 S^{4}} \sum_{cyclic} a^{2} S_{A}^{2} (S^{2} - S_{B} S_{C}) - 2 (S_{ω} - 4 R^{2}) + (S_{ω} - 4 R^{2}) \\ = \frac{1}{4 S^{2}} \sum_{cyclic} a^{2} S_{A}^{2} - \frac{S_{A} S_{B} S_{C}}{S^{4}} \sum_{cyclic} a^{2} S_{A} - (S_{ω} - 4 R^{2}) \\ = \frac{1}{4 S^{2}} \sum_{cyclic} a^{2} (b^{2} c^{2} - S^{2}) - \frac{1}{2} (S_{ω} - 4 R^{2}) - (S_{ω} - 4 R^{2}) \\ = \frac{3 a^{2} b^{2} c^{2}}{4 S^{2}} - \frac{1}{4} \sum_{cyclic} a^{2} - \frac{3}{2} (S_{ω} - 4 R^{2}) \\ = 3 R^{2} - \frac{1}{2} S_{ω} - \frac{3}{2} S_{ω} + 6 R^{2} \\ = 9 R^{2} - 2 S_{ω} . \end{matrix}$

Thus,

$| O H | = \sqrt{9 R^{2} - 2 S_{ω}} .$

References

↑ Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. Mathematical Association of America. p. 132. ISBN 978-0883858394.
↑ Loriga, Juan Jacobo Durán, "Nota sobre el triángulo", en El Progreso Matemático, tomo IV (1894), pages 313-316., Periodico de Matematicas Puras y Aplicadas.
↑ Yiu, Paul (2002), "Notation." §3.4.1 in Introduction to the Geometry of the Triangle. pp. 33-34, Version 2.0402, April 2002 (PDF), Department of Mathematics Florida Atlantic University, pp. 33–34.
↑ Kimberling, Clark, Encyclopedia of Triangle Centers - ETC, Part 1 "Introduced on November 1, 2011: Combos" Note 6, University of Evansville.
↑ Yiu, Paul (2002), "The distance formula" §7.1 in Introduction to the Geometry of the Triangle. p. 87, Version 2.0402, April 2002 (PDF), Department of Mathematics Florida Atlantic University, p. 87.
↑ Weisstein, Eric W. "Orthocenter §(14)". MathWorld.

[Chen-2016-1] Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. Mathematical Association of America. p. 132. ISBN 978-0883858394.

[2] Loriga, Juan Jacobo Durán, "Nota sobre el triángulo", en El Progreso Matemático, tomo IV (1894), pages 313-316., Periodico de Matematicas Puras y Aplicadas.

[3] Yiu, Paul (2002), "Notation." §3.4.1 in Introduction to the Geometry of the Triangle. pp. 33-34, Version 2.0402, April 2002 (PDF), Department of Mathematics Florida Atlantic University, pp. 33–34.

[4] Kimberling, Clark, Encyclopedia of Triangle Centers - ETC, Part 1 "Introduced on November 1, 2011: Combos" Note 6, University of Evansville.

[5] Yiu, Paul (2002), "The distance formula" §7.1 in Introduction to the Geometry of the Triangle. p. 87, Version 2.0402, April 2002 (PDF), Department of Mathematics Florida Atlantic University, p. 87.

[6] Weisstein, Eric W. "Orthocenter §(14)". MathWorld.