Question

Теорема Вариньона с доказательством

Answer 1

Ну то, что эо паралелограмм - очевидно.
проведи одну диагональ первого 4-угольника. Стороны нового - средние линии получишихся треугольников, значит они параллельны этой диаганали и друг другу.

Ну а с площадью мне на глаз неочевидно.. . На худой конец - выразить аналитически через координаты.

Answer 2

Theorem of Varignon with Proof

The theorem of Varignon, also known as Varignon's theorem or the midline theorem, states that in a quadrilateral, the midpoints of the sides form a parallelogram. This theorem is named after Pierre Varignon, a French mathematician who lived in the 17th and 18th centuries.

Statement of Varignon's Theorem: In a quadrilateral ABCD, let E, F, G, and H be the midpoints of sides AB, BC, CD, and DA, respectively. Then the quadrilateral EFGH is a parallelogram.

Proof: To prove Varignon's theorem, we can use vector algebra. Let's consider the vectors representing the sides of the quadrilateral:

Let a represent the vector AB, b represent the vector BC, c represent the vector CD, and d represent the vector DA.

Now, the midpoints of the sides can be expressed as:

E = (A + B) / 2 F = (B + C) / 2 G = (C + D) / 2 H = (D + A) / 2

To show that EFGH is a parallelogram, we need to prove that the opposite sides are parallel. Let's consider the vectors representing the sides of the parallelogram:

e = F - E = (B + C) / 2 - (A + B) / 2 = (C - A) / 2 = 1/2 * (c - a) f = G - F = (C + D) / 2 - (B + C) / 2 = (D - B) / 2 = 1/2 * (d - b) g = H - G = (D + A) / 2 - (C + D) / 2 = (A - C) / 2 = 1/2 * (a - c) h = E - H = (A + B) / 2 - (D + A) / 2 = (B - D) / 2 = 1/2 * (b - d)

We can see that e, f, g, and h are all equal to 1/2 times the vectors representing the opposite sides of the quadrilateral.

Since the vectors c - a and d - b are parallel, it follows that e, f, g, and h are also parallel. Therefore, the opposite sides of the quadrilateral EFGH are parallel, and EFGH is a parallelogram.

Thus, we have proved Varignon's theorem.