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Triangle is a term used to describe one of the basic and three sides or edges which are Straight Line Segment s. In Euclidean Geometry any three non- Collinear points determine a triangle and a unique Plane , i.e. two dimensional Cartesian Space . TYPES OF TRIANGLES Triangles can be classified according to the relative lengths of their sides:
Triangles can also be classified according to the their internal angles, described below using Degree s of arc.
BASIC FACTS Elementary facts about triangles were presented by Euclid in books 1-4 of his '' Elements '' around 300 BCE . A triangle is a Polygon and a 2- Simplex (see Polytope ). All triangles are two- Dimension al. The angles of a triangle add up to 180 degrees. An Exterior Angle of a triangle (an angle that is adjacent and supplementary to an internal angle) is always equal to the two angles of a triangle that it is not adjacent/supplementary to. Like all Convex polygons, the exterior angles of a triangle add up to 360 degrees. The sum of the lengths of any two sides of a triangle always exceeds the length of the third side. That is the Triangle Inequality . Two triangles are said to be '' Similar '' if and only if the angles of one are equal to the corresponding angles of the other. In this case, the lengths of their corresponding sides are Proportional . This occurs for example when two triangles share an angle and the sides opposite to that angle are parallel. A few basic postulates and theorems about similar triangles: Two triangles are similar if at least 2 corresponding angles are congruent. If two corresponding sides of two triangles are in proportion, and their included angles are congruent, the triangles are similar. If three sides of two triangles are in proportion, the triangles are similar. For two triangles to be congruent, each of their corresponding angles and sides must be congruent (6 total). A few basic postulates and theorems about congruent triangles: SAS Postulate: If two sides and the included angles of two triangles are correspondingly congruent, the two triangles are congruent. SSS Postulate: If every side of two triangles are correspondingly congruent, the triangles are congruent. ASA Postulate: If two angles and the included sides of two triangles are correspondingly congruent, the two triangles are congruent. AAS Theorem: If two angles and any side of two triangles are correspondingly congruent, the two triangles are congruent. Hypotenuse-Leg Theorem: If the hypotenuses and 1 pair of legs of two right triangles are correspondingly congruent, the triangles are congruent. Using right triangles and the concept of similarity, the Trigonometric Function s sine and cosine can be defined. These are functions of an Angle which are investigated in Trigonometry . In Euclidean geometry, the sum of the internal angles of a triangle is equal to 180°. This allows determination of the third angle of any triangle as soon as two angles are known. A central theorem is the Pythagorean 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 : The converse is true: if the lengths of the sides of a triangle satisfy the above equation, then the triangle is a right triangle. Some other facts about right triangles:
For all triangles, angles and sides are related by the Law Of Cosines and Law Of Sines . POINTS, LINES AND CIRCLES ASSOCIATED WITH A TRIANGLE There are hundreds of different constructions that find a special point 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 gives a useful general criterion. In this section just a few of the most commonly-encountered constructions are explained. 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. .]] 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 . .]] 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 Excircle s; 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 . is the center of gravity.]] 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 . 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. 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 Excircle s. 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. COMPUTING THE AREA OF A TRIANGLE Calculating the area of a triangle is an elementary problem encountered often in many different situations. Various approaches exist, depending on what is known about the triangle. What follows is a selection of frequently used formulae for the area of a triangle. Using vectors | |||||||||||||||
| The Area Of Triangle ABC Is Half Of This, Or ''S''  |  Â˝''AB'' Ă— ''AC'' |
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| Rac{1}{2} \sqrt{(\mathbf{AB} \cdot \mathbf{AB})(\mathbf{AC} \cdot \mathbf{AC}) -(\mathbf{AB} \cdot \mathbf{AC})^2} | rac{1}{2} \sqrt{ \mathbf{AB}^2 \mathbf{AC}^2 -(\mathbf{AB} \cdot \mathbf{AC})^2} \, |
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| :<math>S | rac{1}{2}\left\det\begin{pmatrix}x_B & x_C \ y_B & y_C \end{pmatrix}
ight = rac{1}{2}x_B y_C - x_C y_B </math> |
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| :<math>S | rac{1}{2} \left \det\begin{pmatrix}x_A & x_B & x_C \ y_A & y_B & y_C \ 1 & 1 & 1\end{pmatrix}
ight = rac{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 </math> |
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