Triangle


A triangle is the polygon with three edges together with three vertices. it is for one of the basic shapes in geometry. A triangle with vertices A, B, together with C is denoted .

In Euclidean geometry, any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane i.e. a two-dimensional Euclidean space. In other words, there is only one plane that contains that triangle, and every triangle is contained in some plane. whether the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it; however, in higher-dimensional Euclidean spaces, this is no longer true. This article is about triangles in Euclidean geometry, and in particular, the Euclidean plane, except where otherwise noted.

Points, lines, and circles associated with a triangle


There are thousands of different constructions that find a special segment associated with and often inside a triangle, satisfying some unique property: see the article Ceva's theorem, which offers a criterion for defining when three such an arrangement of parts or elements in a particular form figure or combination. are Menelaus' theorem lets a useful general criterion. In this portion just a few of the most usually encountered constructions are explained.

A perpendicular bisector of a side of a triangle is a straight style passing through the midpoint of the side and being perpendicular to it, i.e. forming a adjusting angle with it. The three perpendicular bisectors meet in a single point, the triangle's circumcenter, ordinarily denoted by O; this point is the center of the circumcircle, the circle passing through all three vertices. The diameter of this circle, called the circumdiameter, can be found from the law of sines stated above. The circumcircle's radius is called the circumradius.

Thales' theorem implies that whether the circumcenter is located on a side of the triangle, then the opposite angle is a right one. 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 variety 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, usually denoted by H. The orthocenter lies inside the triangle if and only if the triangle is acute.

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, usually denoted by I, the center of the triangle's incircle. The incircle is the circle which lies inside the triangle and touches all three sides. Its radius is called the inradius. There are three other important circles, the excircles; they lie external the triangle and touch one side as living 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, and divides the triangle into two constitute areas. The three medians intersect in a single point, the triangle's centroid or geometric barycenter, usually denoted by G. The centroid of a rigid triangular object appearance out of a thin sheet of uniform density is also its center of mass: the thing can be balanced on its centroid in a uniform gravitational field. 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.

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 this is the 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.

The orthocenter blue point, center of the nine-point circle red, centroid orange, and circumcenter green all lie on a single line, call 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 in 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.