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

where S = 2 × area of reference triangle and

S φ = S cot ⁡ φ . , {\displaystyle S_{\varphi }=S\cot \varphi .,} ³⁴

Basic formulas

In particular:

S A = S cot ⁡ A = b c cos ⁡ A = b 2 + c 2 − a 2 2 , {\displaystyle S_{A}=S\cot A=bc\cos A={\frac {b^{2}+c^{2}-a^{2}}{2}},} S B = S cot ⁡ B = a c cos ⁡ B = a 2 + c 2 − b 2 2 , {\displaystyle S_{B}=S\cot B=ac\cos B={\frac {a^{2}+c^{2}-b^{2}}{2}},} S C = S cot ⁡ C = a b cos ⁡ C = a 2 + b 2 − c 2 2 , {\displaystyle S_{C}=S\cot C=ab\cos C={\frac {a^{2}+b^{2}-c^{2}}{2}},} S ω = S cot ⁡ ω = a 2 + b 2 + c 2 2 , {\displaystyle S_{\omega }=S\cot \omega ={\frac {a^{2}+b^{2}+c^{2}}{2}},}      where ω , {\displaystyle \omega ,} is the Brocard angle. The law of cosines is used: a 2 = b 2 + c 2 − 2 b c cos ⁡ A {\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos A} . S π 3 = S cot ⁡ π 3 = S 3 3 , {\displaystyle S_{\frac {\pi }{3}}=S\cot {\frac {\pi }{3}}=S{\frac {\sqrt {3}}{3}},} S 2 φ = S φ 2 − S 2 2 S φ S φ 2 = S φ + S φ 2 + S 2 , {\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   φ {\displaystyle \varphi }   where   0 < φ < π , {\displaystyle 0<\varphi <\pi ,} S ϑ + φ = S ϑ S φ − S 2 S ϑ + S φ S ϑ − φ = S ϑ S φ + S 2 S φ − S ϑ , . {\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 S ϑ S φ = S ϑ φ , {\displaystyle S_{\vartheta }S_{\varphi }=S_{\vartheta \varphi },} and S ϑ S φ S ψ = S ϑ φ ψ , . {\displaystyle S_{\vartheta }S_{\varphi }S_{\psi }=S_{\vartheta \varphi \psi },.}

Trigonometric relationships

sin ⁡ A = S b c = S S A 2 + S 2 cos ⁡ A = S A b c = S A S A 2 + S 2 tan ⁡ A = S S A , {\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}}},} a 2 = S B + S C b 2 = S A + S C c 2 = S A + S B . {\displaystyle a^{2}=S_{B}+S_{C}\quad \quad b^{2}=S_{A}+S_{C}\quad \quad c^{2}=S_{A}+S_{B}.}

Important identities

∑ cyclic S A = S A + S B + S C = S ω , {\displaystyle \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 , {\displaystyle 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 , {\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},} 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 , {\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,   s = a + b + c 2 , {\displaystyle s={\frac {a+b+c}{2}},}    and   a + b + c = S r . {\displaystyle a+b+c={\frac {S}{r}}.}

Trigonometric conversions

sin ⁡ A sin ⁡ B sin ⁡ C = S 4 R 2 cos ⁡ A cos ⁡ B cos ⁡ C = S ω − 4 R 2 4 R 2 {\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}}}} ∑ cyclic sin ⁡ A = S 2 R r = s R ∑ cyclic cos ⁡ A = r + R R ∑ cyclic tan ⁡ A = S S ω − 4 R 2 = tan ⁡ A tan ⁡ B tan ⁡ C . {\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.}

Useful formulas

∑ cyclic a 2 S A = a 2 S A + b 2 S B + c 2 S C = 2 S 2 ∑ cyclic a 4 = 2 ( S ω 2 − S 2 ) , {\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}),} ∑ cyclic S A 2 = S ω 2 − 2 S 2 ∑ cyclic S B C = ∑ cyclic S B S C = S 2 ∑ cyclic b 2 c 2 = S ω 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}.}

Applications

Let D be the distance between two points P and Q whose trilinear coordinates are pa : pb : pc and qa : qb : qc. Let Kp = apa + bpb + cpc and let Kq = aqa + bqb + cqc. Then D is given by the formula:

D 2 = ∑ cyclic a 2 S A ( p a K p − q a K q ) 2 , . {\displaystyle D^{2}=\sum _{\text{cyclic}}a^{2}S_{A}\left({\frac {p_{a}}{K_{p}}}-{\frac {q_{a}}{K_{q}}}\right)^{2},.} ⁵

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 pa = aSA and for the orthocenter qa = SBSC/a

K p = ∑ cyclic a 2 S A = 2 S 2 K q = ∑ cyclic S B S C = S 2 , . {\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:

D 2 = ∑ cyclic a 2 S A ( a S A 2 S 2 − S B S C a S 2 ) 2 = 1 4 S 4 ∑ cyclic a 4 S A 3 − S A S B S C S 4 ∑ cyclic a 2 S A + S A S B S C S 4 ∑ cyclic S B S C = 1 4 S 4 ∑ cyclic a 2 S A 2 ( S 2 − S B S C ) − 2 ( S ω − 4 R 2 ) + ( S ω − 4 R 2 ) = 1 4 S 2 ∑ cyclic a 2 S A 2 − S A S B S C S 4 ∑ cyclic a 2 S A − ( S ω − 4 R 2 ) = 1 4 S 2 ∑ cyclic a 2 ( b 2 c 2 − S 2 ) − 1 2 ( S ω − 4 R 2 ) − ( S ω − 4 R 2 ) = 3 a 2 b 2 c 2 4 S 2 − 1 4 ∑ cyclic a 2 − 3 2 ( S ω − 4 R 2 ) = 3 R 2 − 1 2 S ω − 3 2 S ω + 6 R 2 = 9 R 2 − 2 S ω . {\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}}}

Thus,

O H = 9 R 2 − 2 S ω , . {\displaystyle OH={\sqrt {9R^{2}-2S_{\omega },}}.} ⁶

See also

• Brocard angle
• Orthocenter
• Circumcenter
• Trilinear coordinates

References

1. Chen, Evan (2016). Euclidean Geometry in Mathematical Olympiads. Mathematical Association of America. p. 132. ISBN 978-0883858394. 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. https://hemerotecadigital.bne.es/hd/es/viewer?id=60bef4e2-9410-4e51-8dca-5044fc99ba4a ↩

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. https://mathematicalolympiads.wordpress.com/wp-content/uploads/2012/08/geometrynotes.pdf ↩

4. Kimberling, Clark, Encyclopedia of Triangle Centers - ETC, Part 1 "Introduced on November 1, 2011: Combos" Note 6, University of Evansville. https://faculty.evansville.edu/ck6/encyclopedia/ETC.html ↩

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. https://mathematicalolympiads.wordpress.com/wp-content/uploads/2012/08/geometrynotes.pdf ↩

6. Weisstein, Eric W. "Orthocenter §(14)". MathWorld. /wiki/Eric_W._Weisstein ↩