A triangle is a 3-sided Polygon sometimes (but not very commonly) called the Trigon. All triangles are convex. An Acute Triangle is a triangle whose three angles are all Acute. A triangle with all sides equal is called Equilateral. A triangle with two sides equal is called Isosceles. A triangle having an Obtuse Angle is called an Obtuse Triangle. A triangle with a Right Angle is called Right. A triangle with all sides a different length is called Scalene.

The sum of Angles in a triangle is 180°. This can be established as follows. Let (
be Parallel to ) in the above diagram, then the angles and satisfy
and
, as indicated. Adding , it follows that

(1) |

Let stand for a triangle side and for an angle, and let a set of s and s be concatenated such that adjacent letters correspond to adjacent sides and angles in a triangle. Triangles are uniquely determined by specifying three sides (SSS Theorem), two angles and a side (AAS Theorem), or two sides with an adjacent angle (SAS Theorem). In each of these cases, the unknown three quantities (there are three sides and three angles total) can be uniquely determined. Other combinations of sides and angles do not uniquely determine a triangle: three angles specify a triangle only modulo a scale size (AAA Theorem), and one angle and two sides not containing it may specify one, two, or no triangles (ASS Theorem).

The Straightedge and Compass construction of the triangle can be accomplished as follows. In the above figure, take as a Radius and draw . Then bisect and construct . Extending to locate then gives the Equilateral Triangle .

In Proposition IV.4 of the *Elements*, Euclid showed how to inscribe a Circle (the
Incircle) in a given triangle by locating the Center as the point of intersection of Angle
Bisectors. In Proposition IV.5, he showed how to circumscribe a Circle (the Circumcircle)
about a given triangle by locating the Center as the point of intersection of the perpendicular bisectors.

If the coordinates of the triangle Vertices are given by where 2, 3, then
the Area is given by the Determinant

(2) |

(3) |

In the above figure, let the Circumcircle passing through a triangle's Vertices have
Radius , and denote the Central Angles from the first point to the second ,
and to the third point by . Then the Area of the triangle is given by

(4) |

If a triangle has sides , , , call the angles opposite these sides , , and , respectively. Also
define the Semiperimeter as Half the Perimeter:

(5) |

(6) |

(7) | |||

(8) | |||

(9) | |||

(10) | |||

(11) | |||

(12) | |||

(13) | |||

(14) |

In the above formulas, is the Altitude on side , is the Circumradius, and is the Inradius (Johnson 1929, p. 11). Expressing the side lengths , , and in terms of the radii , , and of the mutually tangent circles centered on the Triangle vertices (which define the Soddy Circles),

(15) | |||

(16) | |||

(17) |

gives the particularly pretty form

(18) |

The Angles of a triangle satisfy

(19) |

(20) |

Let a triangle have Angles , , and . Then

(21) |

(22) |

(23) |

Trigonometric Functions of half angles can be expressed in terms of the triangle sides:

(24) | |||

(25) | |||

(26) |

where is the Semiperimeter.

The number of different triangles which have Integral sides and Perimeter is

(27) |

where and are Partition Function

(28) |

It is not known if a triangle with Integer sides, Medians, and Area exists
(although there are incorrect Proofs of the impossibility in the literature). However, R. L. Rathbun,
A. Kemnitz, and R. H. Buchholz have shown that there are infinitely many triangles with Rational sides (Heronian Triangles) with *two* Rational Medians (Guy 1994).

In the following paragraph, assume the specified sides and angles are adjacent to each other. Specifying three
Angles does not uniquely define a triangle, but any two triangles with the same Angles
are similar (the AAA Theorem). Specifying two Angles and and a side uniquely
determines a triangle with Area

(29) |

(30) |

(31) |

(32) |

There are four Circles which are tangent to the sides of a triangle, one internal and the rest external. Their centers are the points of intersection of the Angle Bisectors of the triangle.

Any triangle can be positioned such that its shadow under an orthogonal projection is Equilateral.

**References**

Abi-Khuzam, F. ``Proof of Yff's Conjecture on the Brocard Angle of a Triangle.'' *Elem. Math.* **29**, 141-142, 1974.

Andrews, G. ``A Note on Partitions and Triangles with Integer Sides.'' *Amer. Math. Monthly* **86**, 477, 1979.

Baker, M. ``A Collection of Formulæ for the Area of a Plane Triangle.'' *Ann. Math.* **1**, 134-138, 1884.

Berkhan, G. and Meyer, W. F. ``Neuere Dreiecksgeometrie.'' In *Encyklopaedie der Mathematischen Wissenschaften, Vol. 3AB 10*
(Ed. F. Klein). Leipzig: Teubner, pp. 1173-1276, 1914.

Beyer, W. H. (Ed.) *CRC Standard Mathematical Tables, 28th ed.* Boca Raton, FL: CRC Press, pp. 123-124, 1987.

Coxeter, H. S. M. *Introduction to Geometry, 2nd ed.* New York: Wiley, 1969.

Davis, P. ``The Rise, Fall, and Possible Transfiguration of Triangle Geometry: A Mini-History.'' *Amer. Math. Monthly* **102**,
204-214, 1995.

Eppstein, D. ``Triangles and Simplices.'' http://www.ics.uci.edu/~eppstein/junkyard/triangulation.html.

Feuerbach, K. W.
*Eigenschaften einiger merkwürdingen Punkte des geradlinigen Dreiecks, und mehrerer durch die bestimmten Linien und Figuren.*
Nürnberg, Germany, 1822.

Guy, R. K. ``Triangles with Integer Sides, Medians, and Area.'' §D21
in *Unsolved Problems in Number Theory, 2nd ed.* New York: Springer-Verlag, pp. 188-190, 1994.

Honsberger, R. *Mathematical Gems III.* Washington, DC: Math. Assoc. Amer., pp. 39-47, 1985.

Johnson, R. A. *Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle.* Boston, MA:
Houghton Mifflin, 1929.

Jordan, J. H.; Walch, R.; and Wisner, R. J. ``Triangles with Integer Sides.'' *Amer. Math. Monthly* **86**, 686-689, 1979.

Kimberling, C. ``Central Points and Central Lines in the Plane of a Triangle.'' *Math. Mag.* **67**, 163-187, 1994.

Kimberling, C. ``Triangle Centers and Central Triangles.'' *Congr. Numer.* **129**, 1-295, 1998.

Le Lionnais, F. *Les nombres remarquables.* Paris: Hermann, p. 28, 1983.

Schroeder. *Das Dreieck und seine Beruhungskreise.*

Sloane, N. J. A. Sequence
A005044/M0146
in ``An On-Line Version of the Encyclopedia of Integer Sequences.''
http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S.
*The Encyclopedia of Integer Sequences.* San Diego: Academic Press, 1995.

Vandeghen, A. ``Some Remarks on the Isogonal and Cevian Transforms. Alignments of Remarkable Points of a Triangle.''
*Amer. Math. Monthly* **72**, 1091-1094, 1965.

Weisstein, E. W. ``Plane Geometry.'' Mathematica notebook PlaneGeometry.m.

© 1996-9

1999-05-26