Question

Докажите что сумма длин двух противоположных сторон выпуклого четырехугольника меньше суммы длин его диагоналей

Answer 1

Решение прицеплено в картинке

Answer 2

Understanding the Inequality in a Convex Quadrilateral

To prove that the sum of the lengths of two opposite sides of a convex quadrilateral is less than the sum of the lengths of its diagonals, we can use the triangle inequality theorem and the properties of convex quadrilaterals.

Triangle Inequality Theorem

The triangle inequality theorem states that in any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Properties of Convex Quadrilaterals

In a convex quadrilateral, the sum of the lengths of any two sides is greater than the length of the diagonal between the endpoints of those two sides.

Proof

Let's consider a convex quadrilateral ABCD with sides AB, BC, CD, and AD, and diagonals AC and BD. According to the triangle inequality theorem, in triangle ABC, AB + BC > AC, and in triangle ACD, AD + CD > AC. Similarly, in triangle ABD, AB + AD > BD, and in triangle BCD, BC + CD > BD.

Now, adding the inequalities together, we get: AB + BC + AD + CD > AC + BD

This inequality shows that the sum of the lengths of the two opposite sides (AB + CD) is less than the sum of the lengths of the diagonals (AC + BD).

Therefore, the sum of the lengths of two opposite sides of a convex quadrilateral is indeed less than the sum of the lengths of its diagonals.