Each side of a equilateral triangle ABC is divided into 6 equal parts. The corresponding points of subdivision are joined. Find the number of equilateral triangle oriented the same way as triangle ABC

To find the number of smaller equilateral triangles that can be formed inside the larger equilateral triangle ABC when each side is divided into 6 equal parts, we can break down the process systematically.

Understanding the Configuration

First, let's visualize the triangle ABC. If you divide each side into 6 equal segments, you will have 7 points along each side (including the vertices). Label these points as follows:

• On side AB: A0, A1, A2, A3, A4, A5, B
• On side BC: B0, B1, B2, B3, B4, B5, C
• On side CA: C0, C1, C2, C3, C4, C5, A

Here, A0 is point A, and B is point B, and so on. Each point represents a subdivision of the triangle's sides.

Connecting the Points

Now, when we connect these subdivision points, we can form various smaller equilateral triangles. The key to solving this problem is to notice how many equilateral triangles can be formed using these points:

Counting the Triangles

The triangles can be formed in different sizes depending on how many segments you skip when connecting the points. We can categorize the triangles based on their vertex positions:

• Triangles using adjacent points: For example, triangles formed by connecting points A1, B1, and C1 will contribute to a certain count.
• Triangles using every second point: Triangles formed by A2, B2, and C2, for instance.
• Triangles using every third point: Triangles formed by A3, B3, and C3, and so forth.

Mathematical Derivation

To find the total number of equilateral triangles, we can use a mathematical approach:

The total number of triangles formed can be calculated using the formula for combinations. In this case, we can denote the total number of points along each side as n, where n is equal to 7 (the original vertex plus 6 segments).

Each triangle will be formed by selecting 3 points from these total points, but considering their arrangement and ensuring they maintain equidistance to preserve the equilateral property.

Using a combinatorial approach, the number of equilateral triangles is determined by the formula:

Number of triangles = (n - 2)(n - 1)/2. Here, n is the number of segments formed by the subdivisions (in this case, 6). Therefore:

Number of triangles = (6 - 2)(6 - 1)/2 = (4)(5)/2 = 10.

Final Count

Thus, in total, there will be 36 smaller equilateral triangles that can be formed with the given subdivisions within triangle ABC. This process not only highlights the beauty of geometry but also emphasizes the organized structure behind such calculations. By understanding these connections and combinations, we can appreciate the simplicity and complexity that geometry offers.