Properties
- The medians of the triangle intersect at one point, which is called the centroid, and are divided by this point into two parts in a ratio of 2: 1, counting from the vertex.
- The triangle is divided by three medians into six equal triangles.
- The larger side of the triangle corresponds to the smaller median.
- From the vectors that form the medians, you can make a triangle.
- With affine transformations, the median goes over to the median.
- The median of the triangle divides it into two equal parts.
Formula
- The formula for the median in terms of the sides (deduced through the Stewart theorem or by extending to a parallelogram and using the equality in the parallelogram of the sum of the squares of the sides and the sum of the squares of the diagonals):
- Formula of the side in terms of medians:
If the two medians are perpendicular, then the sum of the squares of the sides on which they are dropped is 5 times the square of the third side.
Mnemonic rule
Median monkey,
which has a keen eye,
jump right in the middle
sides against the top,
where is it now.
Notes (edit)
see also
Links
Wikimedia Foundation. 2010.
See what "Triangle Median" is in other dictionaries:
Median: The median of a triangle in planimetry, the segment connecting the apex of the triangle with the middle of the opposite side in statistics, the median is the population value that divides the ranked data series in half Median (statistics) ... ... Wikipedia
Median: The median of a triangle in planimetry, a segment connecting the apex of the triangle with the middle of the opposite side Median (statistics) quantile 0.5 Median (trace) is the midline of the trace drawn between right and left ... Wikipedia
The triangle and its medians. Median of a triangle is a segment inside a triangle that connects the apex of a triangle with the middle of the opposite side, as well as a straight line containing this segment. Contents 1 Properties 2 Formulas ... Wikipedia
The line that connects the apex of the triangle to the midpoint of its base. A complete dictionary of foreign words that have come into use in the Russian language. Popov M., 1907. median (lat.mediana average) 1) geol. the segment connecting the apex of the triangle with ... ... Dictionary of foreign words of the Russian language
Median (from the Latin mediana middle) in geometry, a segment connecting one of the vertices of a triangle with the middle of the opposite side. Three M. of the triangle intersect at one point, which is sometimes called the "center of gravity" of the triangle, so ... Great Soviet Encyclopedia
A triangle is a straight line (or a segment inside the triangle) that connects the apex of the triangle to the middle of the opposite side. Three M. of the triangle intersect at one point, to paradise is called the center of gravity of the triangle, the centroid, or ... ... Encyclopedia of mathematics
- (from Latin mediana middle) a segment connecting the apex of the triangle with the middle of the opposite side ... Big Encyclopedic Dictionary
MEDIAN, medians, wives (lat.mediana, lit. middle). 1. A straight line drawn from the apex of the triangle to the middle of the opposite side (mat.). 2. In statistics, for a number of many data, a value with the property that the number of data, ... ... Ushakov's Explanatory Dictionary
MEDIAN, s, wives. In mathematics, a straight line segment that connects the apex of a triangle to the midpoint of the opposite side. Ozhegov's Explanatory Dictionary. S.I. Ozhegov, N.Yu. Shvedova. 1949 1992 ... Ozhegov's Explanatory Dictionary
MEDIAN (from Latin mediana middle), a segment connecting the apex of a triangle with the middle of the opposite side ... encyclopedic Dictionary
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When studying any topic of a school course, you can select a certain minimum of tasks, having mastered the methods of solving which, students will be able to solve any problem at the level of program requirements for the topic being studied. I propose to consider the tasks that will allow you to see the relationship of individual topics of the school mathematics course. Therefore, the compiled system of tasks is an effective means of repetition, generalization and systematization of educational material in the course of preparing students for the exam.
To pass the exam, additional information about some of the elements of the triangle will not be superfluous. Consider the properties of the median of the triangle and the problem, in solving which these properties can be used. The proposed tasks implement the principle of level differentiation. All tasks are conventionally divided into levels (the level is indicated in brackets after each task).
Let us recall some properties of the median of the triangle
Property 1. Prove that the median of a triangle ABC drawn from the top A, less than half the sum of the sides AB and AC.
Proof
https://pandia.ru/text/80/187/images/image002_245.gif "alt =" (! LANG: $ \ displaystyle (\ frac (AB + AC) (2)) $" width="90" height="60">.!}
Property 2. The median cuts the triangle into two equal ones.
Proof
Draw from vertex B of triangle ABC the median BD and the height BE..gif "alt =" (! LANG: Area" width="82" height="46">!}
Since the segment BD is the median, then
Q.E.D.
https://pandia.ru/text/80/187/images/image008_96.gif "alt =" (! LANG: Median" align="left" width="196" height="75 src=">!} Property 4. The medians of a triangle divide the triangle into 6 equal triangles.
Proof
Let us prove that the area of each of the six triangles into which the medians divide triangle ABC is equal to the area of triangle ABC. To do this, consider, for example, the triangle AOF and drop the perpendicular AK from the vertex A to the line BF.
Due to property 2,
https://pandia.ru/text/80/187/images/image013_75.gif "alt =" (! LANG: Median" align="left" width="105" height="132 src=">!}
Property 6. The median in a right-angled triangle, drawn from the vertex of the right angle, is equal to half the hypotenuse.
Proof
https://pandia.ru/text/80/187/images/image015_62.gif "alt =" (! LANG: Median" width="273" height="40 src="> что и требовалось доказать.!}
Consequences:1. The center of the circle described about a right-angled triangle lies in the middle of the hypotenuse.
2. If the length of the median in a triangle is equal to half the length of the side to which it is drawn, then this triangle is right-angled.
TASKS
In solving each subsequent problem, proven properties are used.
№1 Topics: Doubling the median. Difficulty: 2+
Signs and properties of a parallelogram Classes: 8.9
Condition
On the continuation of the median AM triangle ABC per point M postponed segment MD equal to AM... Prove that the quadrilateral ABDC- parallelogram.
Solution
Let's use one of the parallelogram features. Diagonals of a quadrilateral ABDC intersect at the point M and divide it in half, so the quadrilateral ABDC- parallelogram.
1. The median splits a triangle into two triangles of the same area.
2. The medians of the triangle intersect at one point, which divides each of them in a ratio of 2: 1, counting from the vertex. This point is called center of gravity triangle.
3. The entire triangle is divided by its medians into six equal triangles.
Properties of the bisectors of a triangle
1. The bisector of an angle is the locus of points equidistant from the sides of this angle.
2. The bisector of the inner corner of the triangle divides the opposite side into segments proportional to the adjacent sides:.
3. The point of intersection of the bisectors of a triangle is the center of the circle inscribed in this triangle.
Triangle elevation properties
1. In a right-angled triangle, the height drawn from the vertex of the right angle splits it into two triangles similar to the original one.
2. In an acute-angled triangle, two of its heights cut off similar 
triangles.
Properties of the midpoint perpendiculars of a triangle
1. Each point of the perpendicular to the segment is equidistant from the ends of this segment. The converse is also true: each point equidistant from the ends of a segment lies on the perpendicular to it.
2. The point of intersection of the perpendiculars to the sides of the triangle is the center of the circle circumscribed about this triangle.
Midline property of a triangle
The middle line of a triangle is parallel to one of its sides and is equal to half of this side.
Similarity of triangles
Two triangles are similar, if one of the following conditions, called signs of similarity:
· Two corners of one triangle are equal to two corners of another triangle;
· The two sides of one triangle are proportional to the two sides of the other triangle, and the angles formed by these sides are equal;
· The three sides of one triangle are respectively proportional to the three sides of the other triangle.
In such triangles, the corresponding lines (heights, medians, bisectors, etc.) are proportional.
Sine theorem
Cosine theorem
a 2= b 2+ c 2- 2bc cos
Area formulas for a triangle
1. Arbitrary triangle
a, b, c - parties; - the angle between the sides a and b; - semi-perimeter; R - the radius of the circumscribed circle; r - radius of the inscribed circle; S - square; h a - height drawn to 
side a.
S = ah a
S = ab sin
S = pr
2. Right triangle
a, b - legs; c - hypotenuse; h c - side elevation c.
S = ch c S = ab
3. Equilateral triangle
Quadrangles
Parallelogram properties 
· Opposite sides are equal;
· Opposite angles are equal;
· The diagonals are halved by the point of intersection;
· The sum of the angles adjacent to one side is 180 °;
The sum of the squares of the diagonals is equal to the sum of the squares of all sides:
d 1 2 + d 2 2 = 2 (a 2 + b 2).
A quadrilateral is a parallelogram if:
1. Its two opposite sides are equal and parallel.
2. Opposite sides are equal in pairs.
3. Opposite angles are equal in pairs.
4. The diagonals are halved by the intersection point.
Trapezoid properties
· Its middle line is parallel to the bases and equal to their half-sum;
· If the trapezoid is isosceles, then its diagonals are equal and the angles at the base are equal;
· If the trapezoid is isosceles, then a circle can be described around it;
· If the sum of the bases is equal to the sum of the sides, then a circle can be inscribed into it.
Rectangle properties
· The diagonals are equal.
A parallelogram is a rectangle if:
1. One of its corners is straight.
2. Its diagonals are equal.
Diamond properties
· All properties of a parallelogram;
· Diagonals are perpendicular;
· The diagonals are the bisectors of its corners.
1. A parallelogram is a rhombus if:
2. Its two adjacent sides are equal.
3. Its diagonals are perpendicular.
4. One of the diagonals is the bisector of its angle.
Square properties
· All corners of the square are straight;
· The diagonals of the square are equal, mutually perpendicular, the intersection point is halved and the corners of the square are halved.
A rectangle is a square if it has any feature of a rhombus.
Basic formulas
1. Arbitrary convex quadrilateral 
d 1,d 2 - diagonals; - the angle between them; S - square.
Gomel scientific-practical conference of schoolchildren on mathematics, its applications and information technologies "Poisk"
Abstract on the topic:
"Triangle medians"
Students:
9 "class state
educational institutions
"Gomel city
Multidisciplinary gymnasium number 14 "
Morozova Elizabeth
Khodosovskaya Alesya
Supervisor-
Mathematics teacher of the highest category
Safonova Alla Viktorovna
Gomel 2009
Introduction
1. Medians of a triangle and their properties
2. Discovery of the German mathematician G. Leibniz
3. Application of medians in mathematical statistics
4. Medians of the tetrahedron
5. Six proofs of the median theorem
Conclusion
List of sources and literature used
Application
Introduction
Geometry starts with a triangle. For two millennia, the triangle has been, as it were, a symbol of geometry, but it is not a symbol. The triangle is an atom of geometry.
The triangle is inexhaustible - its new properties are constantly being discovered. To talk about all its known properties, a volume is needed that is comparable in volume to the volume of the Great Encyclopedia. We want to talk about the median of a triangle and its properties, as well as about the use of medians.
First, remember that the median of a triangle is a segment connecting the vertices of the triangle with the middle of the opposite side. Medians have many properties. But we will consider one property and 6 different proofs of it. The three medians intersect at one point, called the centroid (center of mass), and are divided in a 2: 1 ratio.
There is a median not only of a triangle, but also of a tetrahedron. The segment connecting the vertex of the tetrahedron with the centroid (the point of intersection of the medians) of the opposite face is called the median of the tetrahedron. We will also consider the property of tetrahedron medians.
Medians are used in mathematical statistics. For example, to find the average of a set of numbers.
1. Triangle medians and their properties
As you know, the medians of a triangle are the segments connecting its vertices with the midpoints of opposite sides. All three medians intersect at one point and divide it in a ratio of 1: 2.
The point of intersection of the medians is also the center of gravity of the triangle. If you hang a cardboard triangle at the point of intersection of its medians, then it will be in a state of equilibrium
It is curious that all six triangles, into which each triangle is divided by its medians, have the same area.
The medians of a triangle through its sides are expressed as follows:
, , .If the two medians are perpendicular, then the sum of the squares of the sides on which they are dropped is 5 times the square of the third side.
Let's build a triangle, the sides of which are equal to the medians of this triangle, then the medians of the constructed triangle will be equal to 3/4 of the sides of the original triangle.
This triangle will be called the first, the triangle from its medians - the second, the triangle from the medians of the second - the third, etc. Then triangles with odd numbers (1,3, 5, 7, ...) are similar to each other and triangles with even numbers ( 2, 4, 6, 8, ...) are also similar to each other.
The sum of the squares of the lengths of all medians of a triangle is equal to ¾ the sum of the squares of the lengths of its sides.
2. Discovery of the German mathematician G. Leibniz
Famous German mathematician G. Leibniz discovered a remarkable fact: the sum of the squares of the distances from an arbitrary point of the plane to the vertices of a triangle lying in this plane is equal to the sum of the squares of the distances from the point of intersection of the medians to its vertices, added with three times the square of the distance from the point of intersection of the medians to the selected point.
It follows from this theorem that the point on the plane for which the sum of the squared distances to the vertices of a given triangle is minimal is the point of intersection of the medians of this triangle.
At the same time, the minimum sum of the distances to the vertices of the triangle (and not their squares) will be for the point from which each side of the triangle is visible at an angle of 120 °, if none of the angles of the triangle is greater than 120 ° (Fermat's point), and for the vertex obtuse angle if it is more than 120 °.
From Leibniz's theorem and the previous statement, it is easy to find the distance d from the point of intersection of the medians to the center of the circumscribed circle. Indeed, according to Leibniz's theorem, this distance is equal to the square root of one third of the difference between the sum of the squares of the distances from the center of the circumscribed circle to the vertices of the triangle and the sum
The squares of the distances from the intersection of the medians to the vertices of the triangle. We get that
.Point M the intersection of the medians of the triangle ABC is the only point of the triangle for which the sum of the vectors MA,MBand MC is zero. Point coordinates M(with respect to arbitrary axes) are equal to the arithmetic mean of the corresponding coordinates of the triangle vertices. From these statements one can obtain a proof of the median theorem.
3. Application of medians in mathematical statistics
Medians can be found not only in geometry, but also in mathematical statistics. Let you need to find the average value of a certain set of numbers
, , ..., a p. You can, of course, take the arithmetic mean as the averageBut sometimes this is inconvenient. Let's say that you need to determine the average height of second-graders in Moscow. Interview 100 students at random and record their height. If one of the guys jokingly says that his height is a kilometer, then the arithmetic mean of the written numbers will be too large. It is much better to take as an average median numbers
, ..., a p.Suppose there are an odd number of numbers, and arrange them in a non-decreasing order. The number in the middle is called the median of the set. For example, the median of a set of numbers 1, 2, 5, 30, 1, 1, 2 is 2 (and the arithmetic mean is much larger - it is 6).
4. Medians of the tetrahedron
It turns out that we can talk about medians not only for a triangle, but also for a tetrahedron. The segment connecting the vertex of the tetrahedron with the centroid (the point of intersection of the medians) of the opposite face is called median tetrahedron. Like the medians of a triangle, the medians of a tetrahedron intersect at one point, the center of mass or centroid of the tetrahedron, but the ratio in which they divide at this point is different - 3: 1, counting from the vertices. The same point lies on all the segments connecting the midpoints of the opposite edges of the tetrahedron, its bimedians, and divides them in half. This can be proved, for example, from mechanical considerations, by placing weights of unit mass at each of the four vertices of the tetrahedron.
5. Six proofs of the median theorem
It has long been noticed that getting acquainted with different solutions to one problem is more useful than with the same type of solutions to different problems. One of the theorems admitting, like many other classical theorems of elementary geometry, several instructive proofs, is
Theorem on the medians of a triangle. Medians, B and C of the triangleABCintersect at some point M, and each of them is divided by this point in the ratio 2:1, counting from the top:AM: M= BM: M= CM: M=2. (1)
In all the proofs presented below, except for the sixth, we only establish that median B passes through point M, which divides median A with respect to 2: 1. If in the corresponding reasoning we replace the segment V per segment WITH , then we get that and WITH goes through M. This will prove that all three medians intersect at some point M, moreover AM: M - 2. Since all medians are equal, you can replace A on V or SS 1 hence (1) follows.