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Channel: ИНФОУРОК

инфоурок

школа

видеоуроки

геометрия

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middle line of a triangle

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the middle line of a triangle is called

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a segment connecting the midpoints of its two

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sides; we will

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prove the theorem about the middle line of a

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triangle; the theorem; the

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middle line of a triangle is parallel to

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one of its sides and is equal to half of

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this side; proof let m.n. the

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middle line of the triangle a b c we prove

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that m.n. parallel to the father and mn is equal to

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half of the father, the triangles b m and b to

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the father are similar according to the second criterion of similarity of

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triangles; in fact, the

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angle b they have in common bm relates to b a

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as b.n. relates to the cylinder head and is equal to 1 2 that is,

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as one relates to two because

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m is the middle of b a n is the middle of bc

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therefore angle 1 is equal to angle 2

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because about similar triangles the

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corresponding angles are equal and m

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is related to a c

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as one is related to two of the equality

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angle 1 is equal to angle 2 it follows that mn is

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parallel to the oce, why because angles

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12 are the corresponding angles for

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straight lines m.n. and the father and the secant

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b and therefore these straight lines are m.n.

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and the father is parallel and from 2 the equality

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follows that m n is equal to half of

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c the theorem is proven using this

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theorem we will solve the following problem task 1 to prove

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that the median of a triangle

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intersect at one point which divides

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each median

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in the ratio of two to one counting from the

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vertex the solution

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will consider an arbitrary one triangle a b

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c denote by the letter o the point of intersection of

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its medians a1 and bb1 and draw the middle

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line a1 and b1 of this triangle, segment

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a and one b1 is parallel to side ab.b.

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therefore, angles 1 and 2 as well as angles 3 and 4

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are equal as crosswise angles at the

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intersection of parallel lines

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abe and one b1 by the secant a1 and bb1

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therefore the triangles a both and one

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ob1 are similar in two angles and therefore their

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sides are proportional, that is, ao

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refers to 1 o as if o relates to b1

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o as abe relates to 1 d 1

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but a b is equal to 21 b1 therefore a y is equal to 2 1

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a and b o is equal to 2 b1 o thus . at the

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intersection of medians

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a1 and b1 divides each of them in the ratio of

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two to one counting from the vertex, it is

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similarly proven that the point of

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intersection of medians

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bb1 and cc one divides each of them in the

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ratio of two to one counting from the vertex

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and therefore coincides with point a and

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so on for

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all three The medians of the triangle about the father

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intersect at the point o and are divided by it in the

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ratio of

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two to one, counting from the vertex

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