# SASTriangle

SASTriangle[a,γ,b]

returns a filled triangle with sides of length a and b and angle γ between them.

# Details

• SASTriangle is also known as side-angle-side triangle.
• SASTriangle 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 SASTriangle:
• SASTriangle returns a Triangle with A at the origin, B on the positive axis, and C in the half-plane .
• SASTriangle allows the lengths a and b to be any positive numbers and the angle γ strictly between 0 and .

# Background & Context

• SASTriangle constructs a side-angle-side triangle. In particular, SASTriangle[a,γ,b] represents the Triangle in with vertices , and located at the origin, on the positive axis and in the upper half-plane, respectively, with a and b the lengths of the sides opposite vertices and and γ. By the SAS theorem, the triangle so specified is unique (up to geometric congruence). SASTriangle allows the lengths a and b to be any positive numbers and the angle γ to be a positive value satisfying . The arguments of SASTriangle may be exact or approximate numeric expressions.
• The Triangle objects returned by SASTriangle can be used as 2D graphics primitives or geometric regions.
• SASTriangle is related to a number of other symbols. AASTriangle, ASATriangle and SSSTriangle return two-dimensional triangles constructed using different angle and/or side specifications. SASTriangle is a special case of Triangle, in the sense that SASTriangle[a,γ,b] is equivalent to Triangle[{{0,0},{x,0},{y,z}}] for xSqrt[a^2+b^2-2 a b Cos[γ]], y(b^2-a bCos[γ])/Sqrt[a^2+b^2-2 a b Cos[γ]] and z(a b Sin[γ])/Sqrt[a^2+b^2-2 a b Cos[γ]].

# Examples

open allclose all

## Basic Examples(4)

A triangle with , , and :

An SASTriangle:

Different styles applied to an SASTriangle:

Area and centroid:

## Scope(14)

### Graphics(4)

#### Specification(2)

SASTriangle evaluates to Triangle with one point at the origin and one edge on the axis:

A triangle with symbolic edge length:

Plot them:

#### Styling(2)

Color directives specify the face color:

FaceForm and EdgeForm can be used to specify the styles of the interior and boundary:

### Regions(10)

Embedding dimension is the dimension of the space the triangle lives in:

Geometric dimension is the dimension of the shape itself:

Membership testing:

Conditions for membership:

Area:

Centroid:

Distance from a point to an SASTriangle:

Visualize it:

Signed distance from a point:

Now plot it:

Nearest point:

Visualize it:

A triangle is bounded:

Find its range:

Integrate over an SASTriangle:

Optimize over it:

Solve equations over an SASTriangle:

## Applications(2)

A triangle with two equal sides is an isosceles triangle:

Visualize it:

Find its area:

The circumcircle of an SASTriangle can be found using Circumsphere:

The circumcircle passes through the three corner points:

Find the midpoints for each edge of the triangle:

The perpendicular bisectors are lines from the circumcenter to the midpoints:

## Properties & Relations(2)

SASTriangle is a specialized case of Triangle:

Any SASTriangle can be represented by a Polygon:

## Neat Examples(1)

Varying one angle:

Wolfram Research (2014), SASTriangle, Wolfram Language function, https://reference.wolfram.com/language/ref/SASTriangle.html.

#### Text

Wolfram Research (2014), SASTriangle, Wolfram Language function, https://reference.wolfram.com/language/ref/SASTriangle.html.

#### CMS

Wolfram Language. 2014. "SASTriangle." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/SASTriangle.html.

#### APA

Wolfram Language. (2014). SASTriangle. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/SASTriangle.html

#### BibTeX

@misc{reference.wolfram_2023_sastriangle, author="Wolfram Research", title="{SASTriangle}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/SASTriangle.html}", note=[Accessed: 21-September-2023 ]}

#### BibLaTeX

@online{reference.wolfram_2023_sastriangle, organization={Wolfram Research}, title={SASTriangle}, year={2014}, url={https://reference.wolfram.com/language/ref/SASTriangle.html}, note=[Accessed: 21-September-2023 ]}