Question

Отрезки МР и ЕК пересекаются в их середине О. Докажите, что МЕ параллелен РК.

Answer 1

Строим выпуклый четырёхугольник на концах двух заданных отрезков.
Из свойства пересечения диагоналей и деления в точке пересечения пополам, этот четырёхугольник - параллелограмм. 
Из свойств и определения  параллелограмма - противолежащие стороны параллельны.

Answer 2

Proof that ME is parallel to RK

To prove that ME is parallel to RK, we can use the fact that the segments MR and EK intersect at their midpoint O.

Let's assume that MR and EK intersect at point O.

We can use the midpoint theorem, which states that if a line segment is divided into two equal parts by a point, then the line segment connecting the two endpoints of the divided segment is parallel to the line segment connecting the dividing point to the midpoint of the original segment.

In this case, MR is divided into two equal parts by point O, and EK is also divided into two equal parts by point O.

According to the midpoint theorem, the line segment connecting the endpoints of MR (M and R) is parallel to the line segment connecting O to the midpoint of MR. Similarly, the line segment connecting the endpoints of EK (E and K) is parallel to the line segment connecting O to the midpoint of EK.

Let's denote the midpoint of MR as P and the midpoint of EK as Q.

Therefore, we can conclude that MP is parallel to OR and EQ is parallel to OK.

Now, let's consider the line segments ME and RK.

Since MP is parallel to OR and EQ is parallel to OK, we can apply the transitive property of parallel lines.

According to the transitive property, if two lines are parallel to the same line, then they are parallel to each other.

Therefore, since MP is parallel to OR and EQ is parallel to OK, we can conclude that ME is parallel to RK.

Thus, we have proved that ME is parallel to RK.

Please note that the sources provided do not contain specific information related to the given question. The proof provided is based on the geometric concept of the midpoint theorem and the transitive property of parallel lines.