Triangle

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A triangle.

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted \triangle ABC.

In Euclidean geometry any three non-collinear points determine a unique triangle and a unique plane (i.e. a two-dimensional Euclidean space).

Contents

[edit] Types of triangles

[edit] By relative lengths of sides

Triangles can be classified according to the relative lengths of their sides:

Equilateral Triangle Isosceles triangle Scalene triangle
EquilateralIsoscelesScalene

[edit] By internal angles

Triangles can also be classified according to their internal angles, measured here in degrees.

A triangle that has two angles with the same measure also has two sides with the same length, and therefore it is an isosceles triangle. It follows that in a triangle where all angles have the same measure, all three sides have the same length, and therefore such triangle is equilateral.

Right triangle Obtuse triangle Acute triangle
RightObtuseAcute
 \underbrace{\qquad \qquad \qquad \qquad \qquad \qquad}_{}
 Oblique

[edit] Basic facts

Triangles are assumed to be two-dimensional plane figures, unless the context provides otherwise (see Non-planar triangles, below). In rigorous treatments, a triangle is therefore called a 2-simplex (see also Polytope). Elementary facts about triangles were presented by Euclid in books 1–4 of his Elements, around 300 BC.

The measures of the interior angles of a triangle in Euclidean space always add up to 180 degrees. This allows determination of the measure of the third angle of any triangle as soon as the measure of two angles are known. An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees.[5]

The sum of the lengths of any two sides of a triangle always exceeds the length of the third side, a principle known as the triangle inequality. Since the vertices of a triangle are assumed to be non collinear, it is not possible for the sum of the length of two sides be equal to the length of the third side.

Two triangles that are under a correspondence are said to be similar if every angle of one triangle has the same measure as the corresponding angle in the other triangle and the corresponding sides have lengths that are in the same proportion.

A few basic theorems about similar triangles:

Two triangles that are congruent have exactly the same size and shape:[7] all pairs of corresponding interior angles are equal in measure, and all pairs of corresponding sides have the same length. (This is a total of six equalities, but three are often sufficient to prove congruence.)

Some sufficient conditions for a pair of triangles to be congruent are:

An important case:

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.

The Pythagorean theorem

A central theorem is the theorem, which states in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the two other sides. If the hypotenuse has length c, and the legs have lengths a and b, then the theorem states that

a^2 + b^2 = c^2.\,

The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle has a right angle opposite side c.

Some other facts about right triangles:

a + b + 90^{\circ} = 180^{\circ} \Rightarrow a + b = 90^{\circ} \Rightarrow a = 90^{\circ} - b
c = 2a\,
b = a\times\sqrt{3}

For all triangles, angles and sides are related by the law of cosines and law of sines (also called the cosine rule and sine rule).

[edit] Points, lines and circles associated with a triangle

There are hundreds of different constructions that find a special point associated with (and often inside) a triangle, satisfying some unique property: see the references section for a catalogue of them. Often they are constructed by finding three lines associated in a symmetrical way with the three sides (or vertices) and then proving that the three lines meet in a single point: an important tool for proving the existence of these is Ceva's theorem, which gives a criterion for determining when three such lines are concurrent. Similarly, lines associated with a triangle are often constructed by proving that three symmetrically constructed points are collinear: here Menelaus' theorem gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained.

The circumcenter is the center of a circle passing through the three vertices of the 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 can be found from the law of sines stated above.

Thales' theorem implies 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 obtuse.

The intersection of the altitudes is the orthocenter.

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 acute. The three vertices together with the orthocenter are said to form an orthocentric system.

The intersection of the angle bisectors finds the center of the incircle.

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, the incenter, the center of the triangle's incircle. The incircle is 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.

The intersection of the medians is the centroid.

A median of a triangle is a straight line through a vertex and the midpoint of the opposite side, and divides the triangle into two equal areas. The three medians intersect in a single point, the triangle's centroid. The centroid of a stiff triangular object (cut out of a thin sheet of uniform density) is also its center of gravity: the object can be balanced on its centroid. The centroid cuts every median in the ratio 2:1, i.e. the distance between a vertex and the centroid is twice the distance between the centroid and the midpoint of the opposite side.

Nine-point circle demonstrates a symmetry where six points lie on the edge of the triangle.

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. The remaining three points for which it is named are the midpoints of the portion of altitude between the vertices and the orthocenter. The radius of the nine-point circle is half that of the circumcircle. It touches the incircle (at the Feuerbach point) and the three excircles.

Euler's line is a straight line through the centroid (orange), orthocenter (blue), circumcenter (green) and center of the nine-point circle (red).

The centroid (yellow), orthocenter (blue), circumcenter (green) and barycenter of the nine-point circle (red point) all lie on a single line, known as Euler's line (red 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.

The center of the incircle is not in general located on Euler's line.

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.

[edit] Computing the area of a triangle

The area of a triangle can be demonstrated as half of the area of a paralellogram which has the same base length and height.

Calculating the area of a triangle is an elementary problem encountered often in many different situations. The best known and simplest formula is:

S=\frac{1}{2}bh

where S is area, b is the length of the base of the triangle, and h is the height or altitude of the triangle. The term 'base' denotes any side, and 'height' denotes the length of a perpendicular from the point opposite the side onto the side itself.

Although simple, this formula is only useful if the height can be readily found. For example, the surveyor of a triangular field measures the length of each side, and can find the area from his results without having to construct a 'height'. Various methods may be used in practice, depending on what is known about the triangle. The following is a selection of frequently used formulae for the area of a triangle.[8]

[edit] Using vectors

The area of a parallelogram embedded in a three-dimensional Euclidean space can be calculated using vectors. Let vectors AB and AC point respectively from A to B and from A to C. The area of parallelogram ABDC is then |{AB}\times{AC}|, which is the magnitude of the cross product of vectors AB and AC. |{AB}\times{AC}| is equal to |{h}\times{AC}|, where h represents the altitude h as a vector.

The area of triangle ABC is half of this, or S = \frac{1}{2}|{AB}\times{AC}|.

The area of triangle ABC can also be expressed in terms of dot products as follows:


\frac{1}{2} \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} =\frac{1}{2} \sqrt{ |\mathbf{AB}|^2 |\mathbf{AC}|^2 -(\mathbf{AB} \cdot \mathbf{AC})^2}.\,

In two-dimensional Euclidean space, expressing vector AB as a free vector in Cartesian space equal to (x1,y1) and AC as (x2,y2), this can be rewritten as:


\frac{1}{2}\,|x_1 y_2 - x_2 y_1|.\,
Applying trigonometry to find the altitude h.

[edit] Using trigonometry

The height of a triangle can be found through an application of trigonometry. Using the labelling as in the image on the left, the altitude is h = a sin γ. Substituting this in the formula S = ½bh derived above, the area of the triangle can be expressed as:

S =  \frac{1}{2}ab\sin \gamma = \frac{1}{2}bc\sin \alpha  = \frac{1}{2}ca\sin \beta.

(where α is the interior angle at A, β is the interior angle at B, and γ is the interior angle at C).

Furthermore, since sin α = sin (π - α) = sin (β + γ), and similarly for the other two angles:

S = \frac{1}{2}ab\sin (\alpha+\beta) = \frac{1}{2}bc\sin (\beta+\gamma) = \frac{1}{2}ca\sin (\gamma+\alpha).

[edit] Using coordinates

If vertex A is located at the origin (0, 0) of a Cartesian coordinate system and the coordinates of the other two vertices are given by B = (xByB) and C = (xCyC), then the area S can be computed as ½ times the absolute value of the determinant

S=\frac{1}{2}\left|\det\begin{pmatrix}x_B & x_C \\ y_B & y_C \end{pmatrix}\right| = \frac{1}{2}|x_B y_C - x_C y_B|.

For three general vertices, the equation is:

S=\frac{1}{2} \left| \det\begin{pmatrix}x_A & x_B & x_C \\  y_A & y_B & y_C \\ 1 & 1 & 1\end{pmatrix} \right| = \frac{1}{2} \big| x_A y_C - x_A y_B + x_B y_A - x_B y_C + x_C y_B - x_C y_A \big|
S= \frac{1}{2} \big| (x_C - x_A) (y_B - y_A) - (x_B - x_A) (y_C - y_A) \big|.

In three dimensions, the area of a general triangle {A = (xAyAzA), B = (xByBzB) and C = (xCyCzC)} is the Pythagorean sum of the areas of the respective projections on the three principal planes (i.e. x = 0, y = 0 and z = 0):

S=\frac{1}{2} \sqrt{ \left( \det\begin{pmatrix} x_A & x_B & x_C \\ y_A & y_B & y_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 +
\left( \det\begin{pmatrix} y_A & y_B & y_C \\ z_A & z_B & z_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 +
\left( \det\begin{pmatrix} z_A & z_B & z_C \\ x_A & x_B & x_C \\ 1 & 1 & 1 \end{pmatrix} \right)^2 }.

[edit] Using Heron's formula

The shape of the triangle is determined by the lengths of the sides alone. Therefore the area S also can be derived from the lengths of the sides. By Heron's formula:

S = \sqrt{s(s-a)(s-b)(s-c)}

where s = ½ (a + b + c) is the semiperimeter, or half of the triangle's perimeter.

Three equivalent ways of writing Heron's formula are

 S = \frac{1}{4} \sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4)}
 S = \frac{1}{4} \sqrt{2(a^2b^2+a^2c^2+b^2c^2)-(a^4+b^4+c^4)}
 S = \frac{1}{4} \sqrt{(a+b-c) (a-b+c) (-a+b+c) (a+b+c)}.

The area may also be computed by: S = r × s, where r is the inradius, and s is the semiperimeter.

[edit] Computing the sides and angles

In general, there are various accepted methods of calculating the length of a side or the size of an angle. Whilst certain methods may be suited to calculating values of a right-angled triangle, others may be required in more complex situations.

[edit] Trigonometric ratios in right triangles

A right triangle always includes a 90° (π/2 radians) angle, here labeled C. Angles A and B may vary. Trigonometric functions specify the relationships among side lengths and interior angles of a right triangle.

In right triangles, the trigonometric ratios of sine, cosine and tangent can be used to find unknown angles and the lengths of unknown sides. The sides of the triangle are known as follows:

[edit] Sine, cosine and tangent

The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. In our case

\sin A = \frac {\textrm{opposite}} {\textrm{hypotenuse}} = \frac {a} {h}\,.

Note that this ratio does not depend on the particular right triangle chosen, as long as it contains the angle A, since all those triangles are similar.

The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse. In our case

\cos A = \frac {\textrm{adjacent}} {\textrm{hypotenuse}} = \frac {b} {h}\,.

The tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. In our case

\tan A = \frac {\textrm{opposite}} {\textrm{adjacent}} = \frac {a} {b}\,.

The acronym "SOHCAHTOA" is a useful mnemonic for these ratios.

[edit] Inverse functions

The inverse trigonometric functions can be used to calculate the internal angles for a right angled triangle with the length of any two sides.

Arcsin can be used to calculate an angle from the length of the opposite side and the length of the hypotenuse.

\theta = \arcsin \left( \frac{\text{opposite}}{\text{hypotenuse}} \right)

Arccos can be used to calculate an angle from the length of the adjacent side and the length of the hypontenuse.

\theta = \arccos \left( \frac{\text{adjacent}}{\text{hypotenuse}} \right)

Arctan can be used to calculate an angle from the length of the opposite side and the length of the adjacent side.

\theta = \arctan \left( \frac{\text{opposite}}{\text{adjacent}} \right)

In introductory geometry and trigonometry courses, the notation sin−1, cos−1, etc., are often used in place of arcsin, arccos, etc. However, the arcsin, arccos, etc., notation is standard in higher mathematics where trigonometric functions are commonly raised to powers, as this avoids confusion between multiplicative inverse and compositional inverse.

[edit] The sine and cosine rules

A triangle with sides of length a, b and c and angles of α, β and γ respectively.

The law of sines, or sine rule,[9] states that the ratio of the length of a side to the sine of its corresponding opposite angle is constant, that is

\frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma}.

This ratio is equal to the diameter of the circumscribed circle of the given triangle. Another interpretation of this theorem is that every triangle with angles α, β and γ is similar to a triangle with side lengths equal to sinα, sinβ and sinγ. This triangle can be constructed by first constructing a circle of diameter 1, and inscribing in it two of the angles of the triangle. The length of the sides of that triangle will be sinα, sinβ and sinγ. The side whose length is sinα is opposite to the angle whose measure is α, etc.

The law of cosines, or cosine rule, connects the length of an unknown side of a triangle to the length of the other sides and the angle opposite to the unknown side. As per the law:

For a triangle with length of sides a, b, c and angles of α, β, γ respectively, given two known lengths of a triangle a and b, and the angle between the two known sides γ (or the angle opposite to the unknown side c), to calculate the third side c, the following formula can be used:

c^2\ = a^2 + b^2 - 2ab\cos(\gamma)
b^2\ = a^2 + c^2 - 2ac\cos(\beta)
a^2\ = b^2 + c^2 - 2bc\cos(\alpha)

If the lengths of all three sides of any triangle are known the three angles can be calculated:

\alpha=\arccos\left(\frac{b^2+c^2-a^2}{2bc}\right)
\beta=\arccos\left(\frac{a^2+c^2-b^2}{2ac}\right)
\gamma=\arccos\left(\frac{a^2+b^2-c^2}{2ab}\right)

[edit] Non-planar triangles

A non-planar triangle is a triangle which is not contained in a (flat) plane. Examples of non-planar triangles in non-Euclidean geometries are spherical triangles in spherical geometry and hyperbolic triangles in hyperbolic geometry.

While the internal angles in planar triangles always add up to 180°, a hyperbolic triangle has angles that add up to less than 180°, and a spherical triangle has angles that add up to more than 180°. A hyperbolic triangle can be obtained by drawing on a negatively-curved surface, such as a saddle surface, and a spherical triangle can be obtained by drawing on a positively-curved surface such as a sphere. Thus, if one draws a giant triangle on the surface of the Earth, one will find that the sum of its angles is greater than 180°. It is possible to draw a triangle on a sphere such that each of its internal angles is equal to 90°, adding up to a total of 270°.

[edit] See also

[edit] References

  1. Weisstein, Eric W. Equilateral Triangle. From MathWorld--A Wolfram Web Resource. Accessed 15 June 2010.
  2. Mathematicians have traditionally followed Euclid (Book 1 definition 20) in defining an isosceles triangle as having exactly two sides equal,so that equilateral triangles are excluded; but modern references tend to include equilateral triangles: Wiktionary definition of isosceles triangle, Weisstein, Eric W. Isosceles Triangle. From MathWorld--A Wolfram Web Resource. Accessed 15 June 2010.
  3. Weisstein, Eric W. Scalene Triangle. From MathWorld--A Wolfram Web Resource. Accessed 15 June 2010.
  4. Zeidler, Eberhard (2004). Oxford User's Guide to Mathematics. Oxford University Press. 729. ISBN 978-0-19-850763-5. 
  5. The n external angles of any n-sided convex polygon add up to 360 degrees.
  6. Again, in all cases "mirror images" are also similar.
  7. All pairs of congruent triangles are also similar; but not all pairs of similar triangles are congruent.
  8. Weisstein, Eric W. Triangle Area. From MathWorld--A Wolfram Web Resource. Accessed 15 June 2010.
  9. Prof. David E. Joyce. "The Laws of Cosines and Sines". Clark University. http://www.clarku.edu/~djoyce/trig/laws.html. Retrieved 2008-11-1. 

[edit] External links




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