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Median (geometry)

 

 
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Median (geometry)
 
 
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In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a median of a triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 is a line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
  joining a vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 to the midpoint
Midpoint

The midpoint is the middle Point of a line segment. It is Distance from both endpoints. The formula for determining the midpoint of a segment in the plane, with endpoints and is...
 of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side. The median only bisects the vertex angle from which it is drawn in the case of equilateral
Equilateral

In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths....
 triangles.

three medians are concurrent at a point known as the triangle's centroid
Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure is the intersection of all straight lines that divide into two parts of equal moment about the line....
, or center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 of the triangle.






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Triangle
In geometry
Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers....
, a median of a triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 is a line segment
Line segment

In geometry, a line segment is a part of a line that is bounded by two end Point , and contains every point on the line between its end points....
  joining a vertex
Vertex (geometry)

In geometry, a vertex is a special kind of point which describes the corners or intersections of geometric shapes. Vertices are commonly used in computer graphics to define the corners of surfaces in 3d models, where each such point is given as a vector....
 to the midpoint
Midpoint

The midpoint is the middle Point of a line segment. It is Distance from both endpoints. The formula for determining the midpoint of a segment in the plane, with endpoints and is...
 of the opposing side. Every triangle has exactly three medians; one running from each vertex to the opposite side. The median only bisects the vertex angle from which it is drawn in the case of equilateral
Equilateral

In geometry, an equilateral polygon is a polygon which has all sides of the same length.For instance, an equilateral triangle is a triangle of equal edge lengths....
 triangles.

Point of concurrency

The three medians are concurrent at a point known as the triangle's centroid
Centroid

In geometry, the centroid, geometric center, or barycenter of a plane figure is the intersection of all straight lines that divide into two parts of equal moment about the line....
, or center of mass
Center of mass

The center of mass of a system of wiktionary:Particles is a specific point at which, for many purposes, the system's mass behaves as if it were concentrated....
 of the triangle. Note that this means that the centroid is always in the interior of the triangle. Two-thirds of the length of each median is between the vertex and the centroid, while one-third is between the centroid and the midpoint of the opposite side.

Equal-area division

Each median divides the triangle in half; hence the name. The three medians divide the triangle into six smaller triangles of equal area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
.

Any other lines which divide the area of the triangle into two equal parts do not pass through the centroid.

Proof

Consider a triangle ABC. Let D be the midpoint of , E be the midpoint of , F be the midpoint of , and O be the centroid.

By definition, , thus , where represents the area
Area

Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
 of triangle .

We have:

Thus, and

Since , therefore, . Using the same method, you can show that .

Formula for length

Applying Stewart's theorem
Stewart's theorem

In geometry, Stewart's theorem yields a relation between the lengths of the sides of a triangle and the length of segment from a vertex to a point on the opposite side....
 one gets:

where a is the side of the triangle whose midpoint is the extreme point of median m.

See also

  • Angle bisector
  • Altitude (triangle)
    Altitude (triangle)

    In geometry, an altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side or an extension of the opposite side....
  • Ceva's theorem
    Ceva's theorem

    Ceva's theorem is a well-known theorem in elementary geometry.Given a triangle ABC, and points D, E, and F that lie on lines BC, CA, and AB respectively, the theorem states that...


External links

  • at cut-the-knot
    Cut-the-knot

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • at cut-the-knot
    Cut-the-knot

    Cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics....
  • With interactive animation
  • animated demonstration