在平面四边形ABCD中，AB = AC = CD = 1,\angle ADC = 30^{\cir...

Question

在平面四边形ABCD中，AB = AC = CD = 1,\angle ADC = 30^{\circ},\angle DAB = 120^{\circ}。将\triangle ACD沿AC翻折至\triangle ACP，其中P为动点。求二面角A - CP - B的余弦值的最小值

Réflexion

Interpreting problem statement

Okay, I’ve got a quadrilateral ABCD. The key points are that AB = AC = CD = 1, and the angles ∠ADC = 30° and ∠DAB = 120°. Then, triangle ACD folds along AC, creating triangle ACP, with P as a moving point. Now, I need to find the minimum value of the cosine of the dihedral angle A-CP-B. I’ll focus on how the folding affects the dihedral angle and how to optimize it. Let’s break this down step by step!

Breaking down dihedral angle

Okay, I need to figure out the minimum value of cos(∠(A-CP-B)) — this seems to refer to the dihedral angle between two planes. One plane is formed by A, C, and P; the other by C, P, and B. Both planes share the line CP.

The tricky part is interpreting "A-CP-B" as the intersection, and confirming the dihedral angle between these planes, especially since the problem gives specific details about the quadrilateral ABCD. Let’s continue clarifying the geometry!

Reconstructing the geometry

Alright, I’ve got a quadrilateral ABCD. The crucial points are that AB = AC = CD = 1, with ∠ADC = 30° and ∠DAB = 120°. To clarify, A, B, and C are on the unit circle centered at A, so the distances from A to B and A to C are both 1. D, however, is not necessarily on the circle. The goal is to find the minimum value of the cosine of the dihedral angle formed between the planes containing A, C, P and C, P, B. I'll need to carefully consider the folding geometry and the resulting calculations.

None

Interpreting folding and dihedral angle

So after folding triangle ACD over line AC to form triangle ACP, the point D becomes point P. This operation makes the quadrilateral ABCD non-planar because P moves out of the original plane. The dihedral angle "A - CP - B" refers to the angle between two planes that share the edge CP: one plane is formed by triangle ACP (the folded triangle), and the other by triangle BCP (which is part of the original quadrilateral). We're tasked with finding the minimum value of the cosine of this dihedral angle as P moves with the folding angle.

Calculating the dihedral angle

I need to find the minimum value of cos(φ), where φ is the dihedral angle between the planes formed by triangle ACP and triangle BCP. Both planes share the edge CP, with point P moving as the folding angle changes.

To compute the dihedral angle, I’ll find the normals of each plane and then use them to calculate the cosine of the angle between the planes. This involves projecting vectors along CP and ultimately minimizing the cosine of the resulting dihedral angle. The key idea is to define P’s position through rotation and solve for this angle.

Considering projections and normals

I'm calculating the dihedral angle between two planes formed by triangles ACP and BCP. To do this, I compute projections of vectors CA and CB onto the plane perpendicular to CP. Afterward, I need to calculate the angle between these projections. The fold angle (t) determines how point D rotates to P, and the dihedral angle changes accordingly. I need to find the cosine of this angle, either through projections or by working with the normals of the corresponding planes and minimizing the cosine value with respect to t.

Minimizing the dihedral angle

To compute the dihedral angle, I calculate the dot product of the projection vectors $v_A$ and $v_B$ . After simplifying $v_A$ and $v_B$ , I need to find their magnitudes and their dot product. Then, I can compute the cosine of the dihedral angle and minimize it with respect to the folding angle $t$ (which varies within the range $0 \leq t \leq \pi$ ). The goal is to minimize the cosine of the angle to find the minimum dihedral angle from the folding process.

None

Reasoned about dihedral angle cosine value for 3m 6s

Réponse

我们将问题简化为下面的几何模型。已知平面四边形 $ABCD$ 满足

$AB=AC=1$ ，
$CD=1$ ，
$\angle ADC=30^\circ$ ，
$\angle DAB=120^\circ$ 。

我们把 $AC$ 作为对角线，并将 $△ACD$ “沿 $AC$ 翻折”（即绕 $AC$ 作刚体旋转）得到新三角形 $ACP$ （其中 $P$ 为 $D$ 翻折后的点，随折叠角变化而动）。其余点（例如 $B$ ）均固定在原平面上。问题要求求出沿公共边 $CP$ 形成的两个平面（一个是折叠后包含 $A,C,P$ 的平面，另一个是含 $B,C,P$ 的平面）所形成的二面角的余弦值的最小值。

下面介绍一种思路（注意：下面的计算可证明，经过化简，问题的最优值为）

\min\cos\theta=\frac{3+\sqrt3}{3\sqrt{1+\sqrt3}},

其中 $\theta$ 就是沿边 $CP$ 的二面角。

【大致思路概述】

坐标系选取与基本数据确定。
选取原平面为 $xy$ ‐平面，令点
$A=(0,0,0),\quad C=(1,0,0).$
由于 $AB=1$ 且 $\angle DAB=120^\circ$ （而经计算可证明 $AD$ 与 $x$ 轴夹角为 $30^\circ$ ），可以取
$B=(-\tfrac{\sqrt3}{2},\,\tfrac{1}{2},\,0).$
同时利用 $CD=1$ 与 $\angle ADC=30^\circ$ （以及几何推理或利用参数法），可得
$D=\Bigl(\tfrac{3}{2},\,\tfrac{\sqrt3}{2},\,0\Bigr).$
折叠操作与 $P$ 的坐标表达。
将 $△ACD$ 关于 $AC$ 旋转（翻折）一个角度 $t$ （当 $t=0$ 时，纸未折； $t=\pi$ 时完全翻面），可得到
$P=\Bigl(1.5,\;\tfrac{\sqrt3}{2}\cos t,\; \tfrac{\sqrt3}{2}\sin t\Bigr).$
（此处利用了绕 $x$ 轴旋转的公式。）
二面角的求法。
考察“二面角 $A\text{–}CP\text{–}B$ ”——即在公共棱 $CP$ 上，由平面 $ACP$ （折叠后那一面）和平面 $BCP$ （包含原平面上固定点 $B$ ）所构成的角。利用经典方法：
- 选取平面 $ACP$ 中经 $C$ 且垂直于 $CP$ 的方向（例如取 $CA$ 投影到垂直 $CP$ 的方向），记为向量 $\mathbf{v}_A$ ；
- 选取平面 $BCP$ 中经 $C$ 且垂直于 $CP$ 的方向（如 $CB$ 的投影），记为 $\mathbf{v}_B$ ；
  则二面角 $\theta$ 的余弦可写为
$\cos\theta=\frac{\mathbf{v}_A\cdot\mathbf{v}_B}{\|\mathbf{v}_A\|\;\|\mathbf{v}_B\|}.$
经过（略去繁琐但可证明的）代数推导，利用已知各边及角的关系，可以把 $\cos\theta$ 表示为折叠角 $t$ （或其余弦 $u=\cos t$ ）的函数。
求极值。
证明可以得到当 $u=\cos t=\frac{3-2\sqrt3}{3}$ （即 $t\approx99^\circ$ ）时，上述函数取极值。经过化简，其极值即为
$\min\cos\theta=\frac{3+\sqrt3}{3\sqrt{1+\sqrt3}}.$
（注意：当 $t=0$ （原始平面时）或 $t=\pi$ （完全翻折）时，二面角余弦均为 1；因此极小值出现在折叠过程中。）

【结论】

因此，二面角 $A\text{–}CP\text{–}B$ 的余弦值的最小值为

\boxed{\frac{3+\sqrt3}{3\sqrt{1+\sqrt3}}}.

这个答案经过化简后就是问题要求的极值。

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