# ASATriangle

ASATriangle[α,c,β]

returns a filled triangle with angles α and β and side length c, and c is adjacent to both angles.

# Details

• ASATriangle is also known as angle-side-angle triangle.
• ASATriangle can be used as a primitive in 2D graphics and as a geometric region in 2D.
• The given (blue) and computed (red) parameters for an ASATriangle:
• ASATriangle returns a Triangle with at the origin, on the positive axis, and in the half-plane .
• ASATriangle allows the length c to be any positive number and the angles α and β to be positive and such that α+β<π.

# Background & Context

• ASATriangle constructs an angle-side-angle triangle. In particular, ASATriangle[α,c,β] represents the Triangle in with vertices , and located at the origin, on the positive axis and in the upper half-plane, respectively, with αBAC, βABC and c the length of the side opposite . By the ASA theorem, the triangle so specified is unique (up to geometric congruence). ASATriangle allows the length c to be any positive number and the angles α and β to be positive values satisfying α+β<π. The arguments of ASATriangle may be exact or approximate numeric expressions.
• The Triangle objects returned by ASATriangle can be used as 2D graphics primitives or geometric regions.
• ASATriangle is related to a number of other symbols. AASTriangle, SASTriangle and SSSTriangle return two-dimensional triangles constructed using different angle and/or side specifications. ASATriangle is a special case of Triangle, in the sense that ASATriangle[α,c,β] is equivalent to Triangle[{{0,0},{c,0},{c x,c y}}] for xCos[α] Csc[α+β] Sin[β] and yCsc[α+β] Sin[α] Sin[β].

# Examples

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## Basic Examples(4)

A triangle with , , and :

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An ASATriangle:

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Different styles applied to ASATriangle:

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Area and centroid:

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Centroid:

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