Median of a triangle. Theorems related to medians of a triangle

To find the median using the sides of a triangle, you do not need to remember an additional formula. It is enough to know the solution algorithm.

First, let's look at the problem in general form.

Given a triangle with sides a, b, c. Find the length of the median drawn to side b.

AB=a, AC=b, BC=c.

On ray BF we plot the segment FD, FD=BF.

Let's connect point D with points A and C.

Quadrilateral ABCD is a parallelogram (by attribute), since its diagonals at the intersection point are divided in half.

Property of the diagonals of a parallelogram: the sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of its sides.

Hence: AC²+BD²=2(AB²+BC²), which means b²+BD²=2(a²+c²),

BD²=2(a²+c²)-b². By construction, BF is half of BD, therefore

This is the formula for finding the median of a triangle based on its sides. It is usually written like this:

Let's move on to consider a specific task.

The sides of the triangle are 13 cm, 14 cm and 15 cm. Find the median of the triangle drawn to its side of average length.

Applying similar reasoning, we get:

AC²+BD²=2(AB²+BC²).

14²+BD²=2(13²+15²)

A median is a segment drawn from the vertex of a triangle to the middle of the opposite side, that is, it divides it in half at the point of intersection. The point at which the median intersects the side opposite the vertex from which it emerges is called the base. Each median of the triangle passes through one point, called the intersection point. The formula for its length can be expressed in several ways.

Formulas for expressing the length of the median

• Often in geometry problems, students have to deal with a segment such as the median of a triangle. The formula for its length is expressed in terms of sides:

where a, b and c are the sides. Moreover, c is the side on which the median falls. This is how the simplest formula looks like. Medians of a triangle are sometimes required for auxiliary calculations. There are other formulas.

• If during the calculation two sides of a triangle and a certain angle α located between them are known, then the length of the median of the triangle, lowered to the third side, will be expressed as follows.

Basic properties

• All medians have one common point of intersection O and are divided by it in a ratio of two to one, if counted from the vertex. This point is called the center of gravity of the triangle.
• The median divides the triangle into two others whose areas are equal. Such triangles are called equal-area.
• If you draw all the medians, the triangle will be divided into 6 equal figures, which will also be triangles.
• If all three sides of a triangle are equal, then each of the medians will also be an altitude and a bisector, that is, perpendicular to the side to which it is drawn, and bisects the angle from which it emerges.
• In an isosceles triangle, the median drawn from the vertex that is opposite the side that is not equal to any other will also be the altitude and bisector. The medians dropped from other vertices are equal. This is also a necessary and sufficient condition for isosceles.
• If a triangle is the base of a regular pyramid, then the height dropped to this base is projected to the point of intersection of all medians.

• In a right triangle, the median drawn to the longest side is equal to half its length.
• Let O be the intersection point of the triangle's medians. The formula below will be true for any point M.

• The median of a triangle has another property. The formula for the square of its length through the squares of the sides is presented below.

Properties of the sides to which the median is drawn

• If you connect any two points of intersection of the medians with the sides on which they are dropped, then the resulting segment will be the midline of the triangle and be one half of the side of the triangle with which it does not have common points.
• The bases of the altitudes and medians in a triangle, as well as the midpoints of the segments connecting the vertices of the triangle with the point of intersection of the altitudes, lie on the same circle.

In conclusion, it is logical to say that one of the most important segments is the median of the triangle. Its formula can be used to find the lengths of its other sides.

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This page is dedicated to a fairly common information resource - the description and calculation of the area of ​​an arbitrary triangle. The difference from other resources is the calculation of area online, directly in the process of reading the article

Area through height and base

This is the easiest formula to remember. In words, this formula sounds like this: The area of ​​a triangle is equal to half the product of the base of the triangle and its height.

In the case of a right triangle, this expression takes on an even simpler meaning: The area of ​​a right triangle is equal to half the product of two sides

area through the sides of the triangle

The area of ​​a triangle expressed through its sides has been known for a very long time - it appears in books dating back to the 1st century BC.

This formula can be expressed in different ways; fortunately, the formulas for calculating the parameters of a triangle are sufficient.

But if you try to think in terms of the times before our era, when there were no formulas in the modern representation, there were no variables and root signs, then the only axiom on the basis of which Heron created his formula was the Pythagorean theorem. And since in those days, irrational numbers were not yet known, and scientists had a rather skeptical view of negative numbers, then whole numbers were used for thinking.

The proof itself will not be here; Heron just assumed that he completed an arbitrary Pythagorean triangle to a rectangle, calculated its area, and divided it by two.

Area via vertex coordinates

When the coordinates of the vertices of the triangle are known, the area formula can be expressed as follows:

The third-order determinant is easily expanded, and therefore calculating the area even in manual mode will not cause any difficulties.

Area across two sides and the angle between them

Area through a side and two angles

This is a rare task, but we also calculated a formula for such initial data. An attentive reader immediately sees the “error”. The title says that area is determined through a side and two angles, that is, through three variables, and all four are present in the formula. How so?

In fact, there is no error, knowing one of the basic axioms of the triangle, which says that the sum of the interior angles of a triangle is always (!!) equal to 180 degrees

Therefore, there is nothing difficult, knowing two angles of a triangle, to find out the third.

Area through the medians of a triangle

It's a beautiful formula, isn't it?

Containing this segment. The point of intersection of the median with the side of the triangle is called base of the median.

• You can also introduce the concept outer median triangle.

Encyclopedic YouTube

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✪ MEDIANS of bisectors and ALTITUDES of a triangle - grade 7

✪ Median of a triangle. Construction. Properties.

✪ bisector, median, altitude of a triangle. Geometry 7th grade

Subtitles

Properties

Main property

All three medians of a triangle intersect at one point, which is called the centroid or center of gravity of the triangle, and are divided by this point into two parts in a ratio of 2:1, counting from the vertex.

Properties of medians of an isosceles triangle

• In an isosceles triangle, two medians drawn to equal sides of the triangle are equal, and the third median is both a bisector and an altitude.
• The converse is also true: if two medians in a triangle are equal, then the triangle is isosceles, and the third median is both a bisector and the altitude of the angle at its vertex.
• In an equilateral triangle, all three medians are equal.

Properties of median bases

• Euler's theorem for a circle of nine points: the bases of the three altitudes of an arbitrary triangle, the midpoints of its three sides ( bases of its medians) and the midpoints of three segments connecting its vertices with the orthocenter, all lie on the same circle (the so-called nine point circle).
• A segment drawn through grounds any two medians of a triangle is its midline. The middle line of a triangle is always parallel to the side of the triangle with which it has no common points.
  • Corollary (Thales' theorem about parallel segments). The midline of a triangle is equal to half the length of the side of the triangle to which it is parallel.

Other properties

• If a triangle versatile (scalene), then its bisector drawn from any vertex lies between the median and height drawn from the same vertex.
• The median divides the triangle into two equal (in area) triangles.
• A triangle is divided by three medians into six equal triangles.
• From the segments forming the medians, you can make a triangle, the area of ​​which will be equal to 3/4 of the entire triangle. The median lengths satisfy the triangle inequality.
• In a right triangle, the median drawn from the vertex with the right angle is equal to half the hypotenuse.
• The larger side of the triangle corresponds to the smaller median.
• Straight segment, symmetrical or isogonally conjugate the internal median relative to the internal bisector is called the symmedian of the triangle. Three simedians pass through one point - Lemoine's point.
• Median of a triangle angle isotomically conjugated to myself.

Basic relationships

In particular, the sum of the squares of the medians of an arbitrary triangle is 3/4 of the sum of the squares of its sides: m a 2 + m b 2 + m c 2 = 3 4 (a 2 + b 2 + c 2) (\displaystyle m_(a)^(2)+m_(b)^(2)+m_(c)^(2) =(\frac (3)(4))(a^(2)+b^(2)+c^(2))).

• Conversely, you can express the length of an arbitrary side of a triangle in terms of medians: