The area of triangles bisected by the center line of each side of a triangle is equal, and the ratio is 2:1. How to prove that we need the help of the great God

The area of triangles bisected by the center line of each side of a triangle is equal, and the ratio is 2:1. How to prove that we need the help of the great God

If the area of the original triangle is S1 = 1 / 2ah, then the area of the separated triangle is S2 = 1 / 2A * 1 / 2h, S1: S2 = 1:2, that is, the area of the triangles equally divided by the center line of each side of the triangle is equal, and the proportion is 2:1

A middle line of a triangle divides the area of a triangle into two equal parts?
And point out its true or false?

Inverse: the line that divides the area of a triangle into two equal parts is a middle line of the triangle, which is obviously a false proposition

Proof: the area of a triangle with three sides of the middle line as the side is 34

It is known that ad, CF and be are the three middle lines of △ ABC. Prove that the area of triangle with AD, CF and be as sides = 34S △ ABC. Prove that as shown in the figure, ad, be and CF are the three middle lines of △ ABC, extend ad to g, make DG = ad, connect BG and GC, take the middle point h of BG, connect FH and CH, and the quadrangle ABGC is parallelogram

Know how to calculate the area of the coordinates of the three vertices of a triangle

Regardless of the vertex position of a triangle, △ PMN can always be expressed by the sum and difference of the areas of a right angled trapezoid (or rectangle) and two right angled triangles. In a right angled coordinate system, it is easier to find the areas of the vertices of a right angled trapezoid and a right angled triangle when the coordinates of their vertices are known

How to prove: the area of three triangles composed of the center of gravity and three vertices of triangle is equal
Just think about it

The center of gravity is the intersection of the three middle lines of a triangle. The distance to the vertex is twice that to the midpoint of the opposite side

The sum of the squares of the distances from the center of gravity to the three vertices of the triangle is the minimum. We must use vector proof! Proof!
Must use the vector proof method! Seek the proof! Do not use the analytic geometry method!

Let G be the center of gravity of the triangle ABC, and p be any point on the plane, so let G be the center of gravity of the triangle ABC, and let p be the center of any point on the plane, and let p be the center of any point in the plane, and let p be the center of any point in the plane, then let p be the center of the triangle ABC, and let p be the center of any point on the plane, and let p be any point on the plane, and let p be a point in the plane, then PA 124; | ^ 2 + | | | | \124; \\| ^ 2 + | | | | | | 124; | \\\\\\\\\\\\124| 2 + | GC | ^ 2) + 2PG * (GA + GB + GC) = 3 | PG ^

A right angle trapezoid, if the bottom is shortened by 5m, the area is reduced by 10 m2, and becomes a square, what is the area of the original trapezoid

Let the upper bottom be a, the lower bottom be a, and the height be H
The trapezoidal area is S1 = (a + b) H / 2
Square area S2 = a ^ 2
b=a+5
S1-S2=10
Substituting a = h
a=h=4m
b=9m
S1=26m^2
That is, the original trapezoid area is 26 square meters

A trapezoid, after its bottom is shortened by 5 cm, becomes a square, and the area is reduced by 20 square cm. What is the area of the original trapezoid?

20 × 2 / 5 = 8 (CM), (8 + 5 + 8) × 8 / 2, = 21 × 8 / 2, = 84 (square cm); answer: the original trapezoid area is 84 square cm

A trapezoid, whose bottom is shortened by five meters, becomes a square, and its area is reduced by 20 square meters. How many square meters is the original trapezoid area?

20 △ 5 × 2 = 8; (8 + 8 + 5) × 8 × 1 / 2 = 84 square meters
A: the original trapezoid area is 84 square meters

A trapezoid, after its bottom is shortened by 5 cm, becomes a square, and the area is reduced by 20 square cm. What is the area of the original trapezoid?

20 × 2 / 5 = 8 (CM), (8 + 5 + 8) × 8 / 2, = 21 × 8 / 2, = 84 (square cm); answer: the original trapezoid area is 84 square cm