Question

Докажите, что если геометрическая фигура имеет две взаимно перпендикулярные оси симметрии, то она имеет и центр симметрии.

Answer 1

Расположим декартову систему координат в точке пересечения осей симметрии фигуры. Пусть f(х;y) - функция, описывающая границу фигуры. Тогда из симметрии относительно оси Y будет следовать
f(x;y) = f(-x;y)
Из симметрии относительно оси Х  f(-x;y) = f(-x;-y)
Таким образом, f(x;y) = f(-x;-y) - что означает центральную симметрию относительно точки начала координат, т.е.точки пересечения осей симметрии фигуры.

Answer 2

Proof of the Existence of a Center of Symmetry in a Geometric Figure with Two Perpendicular Axes of Symmetry

To prove that if a geometric figure has two mutually perpendicular axes of symmetry, then it also has a center of symmetry, we can consider the following:

1. Definition of Axes of Symmetry: - An axis of symmetry is a line that divides a figure into two mirrored halves. If a figure has an axis of symmetry, then any reflection in that axis maps the figure onto itself.

2. Perpendicular Axes of Symmetry: - When a geometric figure has two axes of symmetry that are perpendicular to each other, it means that the figure can be reflected across both axes to produce the same result.

3. Proof of the Existence of a Center of Symmetry: - If a figure has two perpendicular axes of symmetry, then any point on the figure can be reflected across both axes to produce the same result. This implies that there exists a point, which we can call the center of symmetry, such that any reflection of the figure through this point also results in the same figure.

4. Conclusion: - Therefore, if a geometric figure has two mutually perpendicular axes of symmetry, it must also have a center of symmetry.

This conclusion is based on the fundamental properties of symmetry and reflection in geometry, and it logically follows from the given conditions.