Question

Прямая l пересекает смежные стороны квадрата в различных точках . Докажите , что прямая l лежит в плоскости квадрата .

Answer 1

Если в плоскости лежат 2 точки прямой, то и вся прямая лежит в этой плоскости.

Answer 2

Statement:

Прямая l пересекает смежные стороны квадрата в различных точках. Докажите, что прямая l лежит в плоскости квадрата.

Proof:

To prove that the line l lies in the plane of the square, we need to show that the line l is coplanar with the square.

Let's consider a square with sides AB, BC, CD, and DA. The line l intersects the sides AB and CD at points P and Q, respectively. We need to show that the line l also intersects the sides BC and DA at points R and S, respectively.

Since the line l intersects the sides AB and CD, we can draw two line segments from points P and Q perpendicular to the sides BC and DA, respectively. Let's call these line segments PR and QS.

Now, we have two pairs of parallel line segments: AB and CD, and PR and QS. Since the opposite sides of a square are parallel, we can conclude that PR is parallel to AB and QS is parallel to CD.

Since PR is parallel to AB and QS is parallel to CD, and AB and CD are parallel to each other, we can conclude that PR is parallel to QS.

Therefore, the line segments PR and QS are parallel to each other. This means that the line l intersects the sides BC and DA at points R and S, respectively.

Since the line l intersects all four sides of the square, it lies in the same plane as the square.

Thus, we have proved that the line l lies in the plane of the square.

Note: The proof provided above is a logical deduction based on the given information. It does not rely on external sources.