Question

через основание равнобедренного треугольник проведена плоскость альфа , докажите что прямая , проходящая через середины боковых сторон этого треугольника параллельна плоскости альфа

Answer 1

Средняя линия треугольника параллельна основанию, через которое проведена плоскость альфа. Значит она параллельна прямой, принадлежащей плоскости альфа, значит параллельна плоскости альфа.

Утверждение справедливо для любых треугольников, не только равнобедренных. 

Answer 2

Proof that the line passing through the midpoints of the legs of an isosceles triangle is parallel to the plane alpha

To prove that the line passing through the midpoints of the legs of an isosceles triangle is parallel to the plane alpha, we can use the concept of parallel lines and the properties of isosceles triangles.

Let's consider an isosceles triangle ABC, where AB = AC. The line passing through the midpoints of the legs of this triangle is the line segment connecting the midpoints of AB and AC. Let's call these midpoints M and N, respectively.

To prove that MN is parallel to the plane alpha, we need to show that the line MN does not intersect the plane alpha, or in other words, the line MN lies entirely in a plane parallel to alpha.

To do this, we can use the fact that the line MN is a midsegment of the triangle ABC. The midsegment of a triangle is a line segment that connects the midpoints of two sides of the triangle. In an isosceles triangle, the midsegment is parallel to the base and half its length.

Since MN is a midsegment of the isosceles triangle ABC, it is parallel to the base BC and half its length. Therefore, MN is parallel to BC, which lies in the plane alpha.

Hence, we have proved that the line passing through the midpoints of the legs of an isosceles triangle is parallel to the plane alpha.