B. Arpa and an exam about geometry

Codeforces

IDCF851B

Time2000ms

Memory256MB

Difficulty—

geometrymath

English · Original

Chinese · Translation

Formal · Original

Arpa is taking a geometry exam. Here is the last problem of the exam. You are given three points _a_, _b_, _c_. Find a point and an angle such that if we rotate the page around the point by the angle, the new position of _a_ is the same as the old position of _b_, and the new position of _b_ is the same as the old position of _c_. Arpa is doubting if the problem has a solution or not (i.e. if there exists a point and an angle satisfying the condition). Help Arpa determine if the question has a solution or not. ## Input The only line contains six integers _a__x_, _a__y_, _b__x_, _b__y_, _c__x_, _c__y_ (|_a__x_|, |_a__y_|, |_b__x_|, |_b__y_|, |_c__x_|, |_c__y_| ≤ 109). It's guaranteed that the points are distinct. ## Output Print "_Yes_" if the problem has a solution, "_No_" otherwise. You can print each letter in any case (upper or lower). [samples] ## Note In the first sample test, rotate the page around (0.5, 0.5) by . In the second sample test, you can't find any solution.

Arpa 正在参加一场几何考试。这是考试的最后一道题。给你三个点 #cf_span[a, b, c]。请找出一个点和一个角度，使得将纸张绕该点旋转该角度后，#cf_span[a] 的新位置与 #cf_span[b] 的原位置相同，且 #cf_span[b] 的新位置与 #cf_span[c] 的原位置相同。 Arpa 怀疑这个问题是否有解（即是否存在满足条件的点和角度）。请帮助 Arpa 判断该问题是否有解。输入仅一行，包含六个整数 #cf_span[ax, ay, bx, by, cx, cy]（#cf_span[|ax|, |ay|, |bx|, |by|, |cx|, |cy| ≤ 10^9]）。保证这三个点互不相同。如果问题有解，请输出 "_Yes_"，否则输出 "_No_"。你可以以任意大小写形式输出每个字母。在第一个样例中，将纸张绕 #cf_span[(0.5, 0.5)] 旋转。在第二个样例中，你无法找到任何解。 ## Input 输入仅一行，包含六个整数 #cf_span[ax, ay, bx, by, cx, cy]（#cf_span[|ax|, |ay|, |bx|, |by|, |cx|, |cy| ≤ 10^9]）。保证这三个点互不相同。 ## Output 如果问题有解，请输出 "_Yes_"，否则输出 "_No_"。你可以以任意大小写形式输出每个字母。 [samples] ## Note 在第一个样例中，将纸张绕 #cf_span[(0.5, 0.5)] 旋转。在第二个样例中，你无法找到任何解。

**Definitions** Let $ A = (a_x, a_y) $, $ B = (b_x, b_y) $, $ C = (c_x, c_y) \in \mathbb{R}^2 $ be three distinct points. **Constraints** $ |a_x|, |a_y|, |b_x|, |b_y|, |c_x|, |c_y| \leq 10^9 $, and $ A \neq B \neq C \neq A $. **Objective** Determine whether there exists a point $ P \in \mathbb{R}^2 $ and an angle $ \theta \in \mathbb{R} $ such that a rotation about $ P $ by $ \theta $ maps: - $ A \mapsto B $, - $ B \mapsto C $. **Condition for Solution** Such a rotation exists if and only if: $$ |A - B| = |B - C| \quad \text{and} \quad A, B, C \text{ are not collinear}. $$

Samples

Input #1

0 1 1 1 1 0

Output #1

Yes

Input #2

1 1 0 0 1000 1000

Output #2

No

API Response (JSON)

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      "statement_type": "Markdown",
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