Question

Prove that the polyhedron whose vertices are centers of faces of a tetrahedron is a tetrahedron again.

Short answer

The polyhedron is a tetrahedron, formed by connecting the centroids of the original tetrahedron's faces.

Step-by-step solution

Understanding the Problem

We start by understanding the problem—determine if the polyhedron formed by connecting the face centers of a tetrahedron is itself a tetrahedron. Recall that a tetrahedron has four triangular faces.

Determine Vertices of the New Polyhedron

The given tetrahedron has four faces. Each face has a center, which is the centroid of the face. So, the polyhedron we form will have four vertices.

Connect the Centers to Form Edges

To form this polyhedron, connect the centroids of adjacent faces. Each face shares an edge with its neighboring faces, suggesting that each center connects to the centers of three adjacent faces.

Establish Geometry of the New Polyhedron

Recognize that if each vertex (center of a face) connects to the three other vertices, we form a closed 3D shape with four triangular faces. This configuration defines a tetrahedron.

Conclude the Structure

Thus, the polyhedron formed by the centers of the faces of an original tetrahedron has exactly the structure of a tetrahedron, confirming that it is indeed a tetrahedron.

Key concepts

Polyhedron

A polyhedron is a 3-dimensional solid in geometry that consists of flat polygonal faces, straight edges, and vertices. Think of it as a box-like shape that is defined by its faces. Each face is typically a polygon, and in a polyhedron, these polygons come together to form a closed surface.

The simplest polyhedron is a tetrahedron, which has exactly four triangular faces. This is the classic pyramid shape, where all vertices lie on a sphere's surface. But polyhedra can be quite complex and come in many varieties, including cubes and dodecahedrons.

When we talk about the polyhedron formed by the centers of the faces of a tetrahedron, we're interested in a new shape that emerges. Each vertex of this new polyhedron corresponds to the centroid of a face from the original tetrahedron. Despite forming the polyhedron from these centroids, the defining property remains: it takes the form of a tetrahedron with four triangular faces.

Centroid

A centroid is a fundamental concept in geometry. It is the "center of mass" or the average position of all the points in a shape. For a triangle, the centroid is the point where the three medians intersect (a median connects a vertex to the midpoint of the opposite side).

When constructing the new polyhedron from a tetrahedron, each triangular face of the original tetrahedron contributes a vertex to this polyhedron through its centroid. This means the new polyhedron’s vertices are the centroids of the original tetrahedron’s faces.

Therefore, knowing how to find the centroid of a triangle is crucial here. To find the centroid of a triangle with vertices at coordinates \(x_1, y_1\), \(x_2, y_2\), and \(x_3, y_3\), use the formula:
\[\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)\]

Geometry

Geometry, the branch of mathematics dealing with shapes and spatial relationships, plays a vital role in understanding polyhedra. It allows us to see how shapes fit together in space.

Understanding the geometry of the new polyhedron involves recognizing its three-dimensional structure. Connecting each centroid to the centroids of surrounding faces forms new edges, which can be visualized as spans that bring the vertices together. This arrangement results in another tetrahedron, showcasing the power of geometric transformations.

Additionally, geometric principles help establish that a polyhedron with four vertices and triangular faces is indeed a tetrahedron. Without geometry, it would be difficult to visualize and prove such concepts.

Triangular Faces

Triangular faces are a key characteristic of a tetrahedron and play a crucial role in the formation of the new polyhedron.

In the original tetrahedron, all faces are triangles. When these faces’ centroids connect, they form a new set of triangular faces. This is because connecting three points (the centroids) naturally creates a triangle.

Hence, every face of the new polyhedron is also a triangle, echoing its origin from a tetrahedron. This symmetrical characteristic of having triangular faces aligns perfectly with the definition of a tetrahedron, verifying that the resultant shape from face centroids is again a tetrahedron.

Understanding why and how these shapes hold their triangular nature is part of what makes geometry both fascinating and essential.