From Wikipedia, the free encyclopedia Median (geometry)
Median (geometry)
Each median divides the area of the triangle in half;
hence the name. (Any other lines which divide the area of
the triangle into two equal parts do not pass through the
centroid.)[1] The three medians divide the triangle into
six smaller triangles of equal area.
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 [ADO] = [BDO],[AFO] = [CFO],[BEO] = [CEO], and
The triangle medians and the centroid. , where [ABC] represents the
area of triangle ; these hold because in each
In geometry, a median of a triangle is a line segment join- case the two triangles have bases of equal length and
ing a vertex to the midpoint of the opposing side. Every share a common altitude from the (extended) base, and a
triangle has exactly three medians: one running from triangle’s area equals one-half its base times its height.
each vertex to the opposite side. In the case of isosceles We have:
and equilateral triangles, a median bisects any angle at a
vertex whose two adjacent sides are equal in length. Other properties
For any triangle,[1]
Relation to center of mass
(perimeter) < sum of the medians < (perimeter).
Each median of a triangle passes through the triangle’s
centroid, which is the center of mass of an object of uni- For any triangle with sides a,b,c and medians ma,mb,mc,[1]
form density in the shape of the triangle. Thus the object
would balance on any line through the centroid, includ-
ing any median.
See also
Equal-area division • Angle bisector
• Altitude (triangle)
References
[1] ^ Posamentier, Alfred S., and Salkind, Charles T.,
Challenging Problems in Geometry, Dover, 1996: pp.
86-87.
External links
• Medians and Area Bisectors of a Triangle
• The Medians at cut-the-knot
• Area of Median Triangle at cut-the-knot
• Medians of a triangle With interactive animation
• Constructing a median of a triangle with compass
and straightedge animated demonstration
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From Wikipedia, the free encyclopedia Median (geometry)
• Weisstein, Eric W., "Triangle Median" from
MathWorld.
Retrieved from "http://en.wikipedia.org/w/index.php?title=Median_(geometry)&oldid=467822174"
Categories:
• Elementary geometry
• Triangles
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