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A triangle is one of the basic shapes of geometry: a two-dimensional figure
with three vertices and three sides which are straight line segments.
Types of triangles
Triangles can be classified according to their side lengths: a triangle is called
* equilateral if all its sides have the same length (or equivalently: all
its angles are equal)
* isosceles if (at least) two of its sides have the same length (or
equivalently: two of its angles are equal)
* scalene if all its sides have different lengths (or equivalently: all
its angles are different)
Triangles can also be classified according to the size of their largest
angle: a triangle is called
* right if one of its angles is a right angle (90 degrees or π/2
radians). The side opposite the right angle is called the hypotenuse.
It is the longest side in the right triangle.
* obtuse if one angle is bigger than a right one
* acute if each angle is smaller than a right one
A triangle is a polygon and a 2-simplex.
Two triangles are said to be similar if one can be produced by uniformly
expanding the other. In this case, the lengths of their sides are
proportional. That is, if the longest side of a triangle is twice that of
the longest side of a similar triangle, say, then the shortest side will
also be twice that of the shortest side of the other triangle, and the
median side will be twice that of the other triangle. Also, the ratio of the
longest side to the shortest in the first triangle will be the same as the
ratio of the longest side to the shortest in the other triangle. The crucial
fact is that two triangles are similar if and only if their corresponding
angles are equal, and this occurs for example when two triangles share an
angle and the sides opposite to that angle are parallel.
Using right triangles and the concept of similarity, the trigonometric
functions sine and cosine can be defined. These are functions of an angle
which are investigated in trigonometry.
In the sequel, we will consider a triangle with vertices A, B and C, angles
α, β and γ and sides a, b and c. The side a is opposite to
the vertex A and angle α and analogously for the other sides.
The sum of the angles α + β + γ is equal to two right
angles (180 degrees or π radians). This allows to determine the third
angle of any triangle as soon as two angles are known.
A central theorem is the Pythagorean theorem stating that in any right
triangle, the square of the hypotenuse is equal to the sum of the squares of
the other two sides. If γ is the right angle, we can write this as
c2 = a2 + b2
This means that knowing the lengths of two sides of a right triangle is
enough to calculate the length of the third -- something unique to right
triangles. The Pythagorean theorem can be generalized to the law of cosines:
c2 = a2 + b2 - 2abcos(γ)
which is valid for all triangles, even if γ is not a right angle. The
law of cosines can be used to compute the side lengths and angles of a
triangle as soon as all three sides or two sides and an enclosed angle are
The law of sines states
sin(α) / a = sin(β) / b = sin(γ) / c
which can be used to compute the side lengths for a triangle as soon as two
angles and one side are known. If two sides and an unenclosed angle is
known, the law of sines may also be used; however, in this case there may be
zero, one or two solutions.
Points, lines and circles associated with a triangle
A perpendicular bisector of a triangle is a straight line passing through
the midpoint of a side and being perpendicular to it, i.e. forming a right
angle with it. The three perpendicular bisectors meet in a single point, the
triangle's circumcenter; this point is the center of the circumcircle, the
circle passing through all three vertices. The diameter of this circle is
given by a/sin(α).
Thales' theorem states that if the circumcenter is located on one side of
the triangle, then the opposite angle is a right one. More is true: if the
circumcenter is located inside the triangle, then the triangle is acute; if
the circumcenter is located outside the triangle, then the triangle is
An altitude of a triangle is a straight line through a vertex and
perpendicular to (i.e. forming a right angle with) the opposite side. This
opposite side is called the base of the altitude, and the point where the
altitude intersects the base (or its extension) is called the foot of the
altitude. The length of the altitude is the distance between the base and
the vertex. The three altitudes intersect in a single point, called the
orthocenter of the triangle. The orthocenter lies inside the triangle if and
only if the triangle is not obtuse. The three vertices together with the
orthocenter are said to form an orthocentric system.
An angle bisector of a triangle is a straight line through a vertex which
cuts the corresponding angle in half. The three angle bisectors intersect in
a single point; this point is the center of the triangle's incircle, the
circle which lies inside the triangle and touches all three sides. There are
three other important circles, the excircles; they lie outside the triangle
and touch one side as well as the extensions of the other two. The centers
of the in- and excircles form an orthocentric system.
A median of a triangle is a straight line through a vertex and the midpoint
of the opposite side. The three medians intersect in a single point, the
triangle's centroid. This is also the triangle's center of gravity: if the
triangle were made out of wood, say, you could balance it on its centroid,
or on any line through the centroid. The centroid cuts every median in the
ratio 2:1, i.e. the distance between a vertex and the centroid is twice as
large as the distance between the centroid and the midpoint of the opposite side.
The midpoints of the three sides and the feet of the three altitudes all lie
on a single circle, the triangle's nine point circle. Its radius is half
that of the circumcircle. It touches the incircle and the three excircles.
The centroid, orthocenter, circumcenter and center of the nine point circle
(but not necessarily the center of the incircle) all lie on a single line,
known as Euler's line. The center of the nine point circle lies at the
midpoint between the orthocenter and the circumcenter, and the distance
between the centroid and the circumcenter is half that between the centroid
and the orthocenter.
If one reflects a median at the angle bisector that passes through the same
vertex, one obtains a symmedian. The three symmedians intersect in a single
point, the symmedian point of the triangle.