Question

Отрезки EF и MN пересекаются в их середине P.Докажите,что EN параллельна MF

Answer 1

Пересекающиеся прямые образуют два равных треугольника т.к. их боковые стороны равны EP=PF=NP=PM по условию задачи и угол EPN= углу MPF как  противолежащие. И так как отрезки EN и MF находятся на одинаковом расстоянии от т. Р и  являются основаниями равных треугольников с противолежащими углами, то они параллельны друг другу.

Answer 2

Proof that EN is parallel to MF

To prove that EN is parallel to MF, we can use the fact that EF and MN intersect at their midpoint P.

Let's assume that EN is not parallel to MF. In that case, EN and MF would intersect at a point other than P. Let's call this point Q.

Since EF and MN intersect at their midpoint P, we can draw a line segment EP and a line segment FP. Similarly, we can draw a line segment MQ and a line segment NQ.

Now, let's consider the triangles EPQ and FPQ. These triangles share the side PQ and have the sides EP and FP parallel to each other. According to the converse of the corresponding angles theorem, if two lines are cut by a transversal and the corresponding angles are congruent, then the lines are parallel.

In this case, EP and FP are parallel, and the corresponding angles EPQ and FPQ are congruent because they are vertical angles. Therefore, by the converse of the corresponding angles theorem, EN and MF must be parallel.

Thus, we have proved that EN is parallel to MF.