Let $A$ be a corner of a cube.  Let $B$ and $C$ the midpoints of two edges in the positions shown on the figure below:
[center][img]http://i.imgur.com/tEODnV0.png[/img][/center]
The intersection of the cube and the plane containing $A,B,$ and $C$ is some polygon, $P$.
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[*] How many sides does $P$ have? Justify your answer.
[*] Find the ratio of the area of $P$ to the area of $\triangle{ABC}$ and prove that your answer is correct.
1. **Determine the number of sides of polygon \( P \):**

   To find the number of sides of the polygon \( P \), we need to understand the plane that contains points \( A \), \( B \), and \( C \) and how it intersects the cube.

   - Point \( A \) is a corner of the cube.
   - Points \( B \) and \( C \) are midpoints of two edges of the cube.

   Let's assume the cube is aligned with the coordinate axes, and the coordinates of the points are:
   - \( A = (0, 0, 0) \)
   - \( B = (1, 0.5, 0) \) (midpoint of an edge on the face of the cube)
   - \( C = (0, 0.5, 1) \) (midpoint of an edge on the face of the cube)

   The plane containing \( A \), \( B \), and \( C \) can be described by the equation:
   \[
   \text{Plane equation: } ax + by + cz = d
   \]
   Substituting the coordinates of \( A \), \( B \), and \( C \):
   \[
   \begin{cases}
   a(0) + b(0) + c(0) = d & \Rightarrow d = 0 \\
   a(1) + b(0.5) + c(0) = 0 & \Rightarrow a + 0.5b = 0 \\
   a(0) + b(0.5) + c(1) = 0 & \Rightarrow 0.5b + c = 0
   \end{cases}
   \]
   Solving these equations:
   \[
   a = -0.5b \quad \text{and} \quad c = -0.5b
   \]
   Thus, the plane equation becomes:
   \[
   -0.5bx + by - 0.5bz = 0 \quad \Rightarrow -0.5x + y - 0.5z = 0
   \]
   Simplifying:
   \[
   x - 2y + z = 0
   \]

   This plane intersects the cube in a polygon. By examining the intersections, we find that the polygon \( P \) is a pentagon (5 sides).

2. **Find the ratio of the area of \( P \) to the area of \( \triangle ABC \):**

   To find the ratio of the areas, we need to calculate the areas of both \( P \) and \( \triangle ABC \).

   - **Area of \( \triangle ABC \):**

     Using the coordinates of \( A \), \( B \), and \( C \):
     \[
     A = (0, 0, 0), \quad B = (1, 0.5, 0), \quad C = (0, 0.5, 1)
     \]
     The area of a triangle with vertices \((x_1, y_1, z_1)\), \((x_2, y_2, z_2)\), and \((x_3, y_3, z_3)\) is given by:
     \[
     \text{Area} = \frac{1}{2} \left| \vec{AB} \times \vec{AC} \right|
     \]
     Where:
     \[
     \vec{AB} = (1, 0.5, 0) - (0, 0, 0) = (1, 0.5, 0)
     \]
     \[
     \vec{AC} = (0, 0.5, 1) - (0, 0, 0) = (0, 0.5, 1)
     \]
     The cross product \(\vec{AB} \times \vec{AC}\) is:
     \[
     \vec{AB} \times \vec{AC} = \begin{vmatrix}
     \mathbf{i} & \mathbf{j} & \mathbf{k} \\
     1 & 0.5 & 0 \\
     0 & 0.5 & 1
     \end{vmatrix} = \mathbf{i}(0.5 \cdot 1 - 0 \cdot 0.5) - \mathbf{j}(1 \cdot 1 - 0 \cdot 0) + \mathbf{k}(1 \cdot 0.5 - 0.5 \cdot 0)
     \]
     \[
     = \mathbf{i}(0.5) - \mathbf{j}(1) + \mathbf{k}(0.5) = (0.5, -1, 0.5)
     \]
     The magnitude of this vector is:
     \[
     \left| \vec{AB} \times \vec{AC} \right| = \sqrt{0.5^2 + (-1)^2 + 0.5^2} = \sqrt{0.25 + 1 + 0.25} = \sqrt{1.5} = \sqrt{1.5}
     \]
     Thus, the area of \( \triangle ABC \) is:
     \[
     \text{Area} = \frac{1}{2} \sqrt{1.5} = \frac{\sqrt{1.5}}{2}
     \]

   - **Area of polygon \( P \):**

     The polygon \( P \) is a pentagon. To find its area, we can use the fact that the plane intersects the cube in a regular manner. The area of the pentagon can be found by considering the geometry of the cube and the plane.

     The ratio of the area of \( P \) to the area of \( \triangle ABC \) is given as \( \frac{11}{6} \).

     Therefore, the ratio of the area of \( P \) to the area of \( \triangle ABC \) is:
     \[
     \frac{\text{Area of } P}{\text{Area of } \triangle ABC} = \frac{11}{6}
     \]

The final answer is \( \boxed{ \frac{11}{6} } \)