Triangle

Triangle

Area of triangle

Medians of triangle

A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side, thus bisecting that side. Every triangle has exactly three medians, one from each vertex, and they all intersect each other at the triangle's centroid.

Altitudes of triangle

The altitude of a triangle is the line drawn from one vertex perpendicular to its opposite side. The altitudes of a triangle intersect at a point called the orthocenter of the triangle.

Angle bisectors of triangle

The angle bisector of an angle of a triangle is a straight line that divides the angle into two congruent angles. The three angle bisectors of the angles of a triangle meet in a single point, called the incenter which is equidistant from the sides of the triangle. The incenter is the center of the inscribed circle of the triangle or incircle.

Perpendicular bisectors of triangle

The perpendicular bisector of a side of a triangle is a line perpendicular to the side and passing through its midpoint. The three perpendicular bisectors of the sides of a triangle meet in a single point, called the circumcenter which is equidistant from the vertices of the triangle. The equal distance is the radius of the circumscribing circle or circumcircle. The circumcircle is the smallest circle that the triangle can be inscribed in.

The circumcenter lies inside the triangle for acute triangles, on the hypotenuse for right triangles and lies outside the triangle for obtuse triangles. The circumcenter coincides with the midpoint of the hypotenuse if it is an isosceles right triangle.

Euler line

The Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral. It is a central line of the triangle, and it passes through several important points determined from the triangle, including the orthocenter (1), the centroid (2), the circumcenter (3), the center of the nine-point circle (4) of the triangle, and the Exeter point.

The nine-point circle is a circle that can be constructed for any given triangle. It is so named because it passes through nine significant concyclic points defined from the triangle. These nine points are, the midpoint of each side of the triangle, the foot of each altitude, the midpoint of the line segment from each vertex of the triangle to the orthocenter.