Plate notation

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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=bcsinA=acsinB=absinC

where S = 2 × area of reference triangle and

Sφ=Scotφ.

in particular

SA=ScotA=bccosA=b2+c2a22
SB=ScotB=accosB=a2+c2b22
SC=ScotC=abcosC=a2+b2c22
Sω=Scotω=a2+b2+c22      where ω is the Brocard angle.
Sπ3=Scotπ3=S33
S2φ=Sφ2S22SφSφ2=Sφ+Sφ2+S2    for values of   φ  where   0<φ<π
Sϑ+φ=SϑSφS2Sϑ+SφSϑφ=SϑSφ+S2SφSϑ

Hence:

sinA=Sbc=SSA2+S2cosA=SAbc=SASA2+S2tanA=SSA

Some important identities:

cyclicSA=SA+SB+SC=Sω
S2=b2c2SA2=a2c2SB2=a2b2SC2
SBSC=S2a2SASASC=S2b2SBSASB=S2c2SC
SASBSC=S2(Sω4R2)Sω=s2r24rR

where R is the circumradius and abc = 2SR and where r is the incenter,   s=a+b+c2   and   a+b+c=Sr

Some useful trigonometric conversions:

sinAsinBsinC=S4R2cosAcosBcosC=Sω4R24R2
cyclicsinA=S2Rr=sRcycliccosA=r+RRcyclictanA=SSω4R2=tanAtanBtanC


Some useful formulas:

cyclica2SA=a2SA+b2SB+c2SC=2S2cyclica4=2(Sω2S2)
cyclicSA2=Sω22S2cyclicSBSC=S2cyclicb2c2=Sω2+S2

Some examples using Conway triangle notation:

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:

D2=cyclica2SA(paKpqaKq)2

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

Kp=cyclica2SA=2S2Kq=cyclicSBSC=S2

Hence:

D2=cyclica2SA(aSA2S2SBSCaS2)2=14S4cyclica4SA3SASBSCS4cyclica2SA+SASBSCS4cyclicSBSC=14S4cyclica2SA2(S2SBSC)2(Sω4R2)+(Sω4R2)=14S2cyclica2SA2SASBSCS4cyclica2SA(Sω4R2)=14S2cyclica2(b2c2S2)12(Sω4R2)(Sω4R2)=3a2b2c24S214cyclica232(Sω4R2)=3R212Sω32Sω+6R2=9R22Sω.

This gives:

OH=9R22Sω.

References

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