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Conway triangle notation

From Wikipedia, the free encyclopedia

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

The notation is named after English mathematician John Horton Conway,[1] who promoted its use, but essentially the same notation (using instead of ) can be found in an 1894 paper by Spanish mathematician Juan Jacobo Durán Loriga [gl].[2]

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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:

where S = 2 × area of reference triangle and

[3][4]
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Basic formulas

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In particular:

     where is the Brocard angle. The law of cosines is used: .
   for values of     where  

Furthermore the convention uses a shorthand notation for and

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Trigonometric relationships

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Important identities

where R is the circumradius and abc = 2SR and where r is the incenter,      and  

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Trigonometric conversions

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Useful formulas

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Applications

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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:

[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

Hence:

Thus,

[6]
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See also

References

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