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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 .[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
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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:
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,
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See also
References
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