In rectangular coordinate system, we often encounter various geometric shapes, such as rectangle, square, circle and so on. However, someone may ask a question: Is there a geometric shape, which is neither a rectangle nor a square, but all four vertices are on the coordinate axis of the coordinate system, that is, forming a diamond? This is the existence of rhombus in rectangular coordinate system.

In order to answer this question, we need to carefully observe the definition of rectangular coordinate system and diamond. Cartesian coordinate system consists of two mutually perpendicular coordinate axes, namely X axis and Y axis. A diamond is a quadrilateral, in which diagonal lines are equal and perpendicular to each other.

First, we can consider the case that all four vertices are on the coordinate axis. Suppose the four vertices of a diamond are A(a, 0), B(0, b), C(-a, 0) and D(0, -b), where A and B are positive real numbers. Through calculation, it can be found that if a and b are not zero, the lengths of diagonal AC and BD are not equal, so the definition of diamond is not satisfied. If one of a and b is zero, the diamond will degenerate into a line segment, which does not conform to the definition of diamond.

Second, we can consider the situation that the vertex of the diamond is not completely on the coordinate axis. In this case, we need to ensure that the diagonals of the diamonds are equal and perpendicular to each other. However, because the coordinate axis of the rectangular coordinate system is infinitely extended and has no fixed unit length, a certain proportion can not be found to satisfy the diagonal equality.

Advantages of learning mathematics well

1. Improve logical thinking ability: Mathematics is a rigorous subject. Learning mathematics can cultivate logical thinking ability and help people to analyze, reason and solve problems better.

2. Enhance problem-solving ability: Mathematics involves methods and skills to solve various problems. Learning mathematics well can cultivate people's problem-solving ability and make people better cope with all kinds of challenges and difficulties.

3. Enhance the ability of abstract thinking: Concepts and theories in mathematics are usually abstract. Learning mathematics well can cultivate people's abstract thinking ability and make people better understand and apply abstract concepts.