Score : $300$ points

问题描述

在 $xy$ 坐标平面上，是否存在一个格点，其到两个格点 $(x_1, y_1)$ 和 $(x_2, y_2)$ 的距离都为 $\sqrt{5}$？

以上为通义千问 qwen-max 翻译，仅供参考。

Problem Statement

On an $xy$-coordinate plane, is there a lattice point whose distances from two lattice points $(x_1, y_1)$ and $(x_2, y_2)$ are both $\sqrt{5}$?

Notes

A point on an $xy$-coordinate plane whose $x$ and $y$ coordinates are both integers is called a lattice point.
The distance between two points $(a, b)$ and $(c, d)$ is defined to be the Euclidean distance between them, $\sqrt{(a - c)^2 + (b-d)^2}$.

The following figure illustrates an $xy$-plane with a black circle at $(0, 0)$ and white circles at the lattice points whose distances from $(0, 0)$ are $\sqrt{5}$. (The grid shows where either $x$ or $y$ is an integer.)

Constraints

• $-10^9 \leq x_1 \leq 10^9$
• $-10^9 \leq y_1 \leq 10^9$
• $-10^9 \leq x_2 \leq 10^9$
• $-10^9 \leq y_2 \leq 10^9$
• $(x_1, y_1) \neq (x_2, y_2)$
• All values in input are integers.

Input

Input is given from Standard Input in the following format:

$x_1$ $y_1$ $x_2$ $y_2$

Output

If there is a lattice point satisfying the condition, print Yes; otherwise, print No.

Sample Input 1

0 0 3 3

Sample Output 1

Yes

• The distance between points $(2,1)$ and $(x_1, y_1)$ is $\sqrt{(0-2)^2 + (0-1)^2} = \sqrt{5}$;
• the distance between points $(2,1)$ and $(x_2, y_2)$ is $\sqrt{(3-2)^2 + (3-1)^2} = \sqrt{5}$;
• point $(2, 1)$ is a lattice point,

so point $(2, 1)$ satisfies the condition. Thus, Yes should be printed.
One can also assert in the same way that $(1, 2)$ also satisfies the condition.

Sample Input 2

0 1 2 3

Sample Output 2

No

No lattice point satisfies the condition, so No should be printed.

Sample Input 3

1000000000 1000000000 999999999 999999999

Sample Output 3

Yes

Point $(10^9 + 1, 10^9 - 2)$ and point $(10^9 - 2, 10^9 + 1)$ satisfy the condition.

update @ 2024/3/10 10:18:54