7. Given three points, the distances between which are 4, 6, and 7. How many pairwise distinct triangles exist for which each of these points is either a vertex or the midpoint of a side?
Answer: 11.

Solution. Let's list all the constructions of triangles that satisfy the condition of the problem, indicating the lengths of the sides.

|  | Description of the Triangle |  | Lengths of Sides |
| :---: | :---: | :---: | :---: |
| №1 | All points are vertices |  | $4,6,7$ |
|  | One point is a vertex, two are midpoints of sides |  |  |
| №2 |  | The vertex is at the extension of both sides of lengths 4 and 6 | $8,12,14$ |
| №3 |  | The extension of the side of length 4 | $8,14,2 \sqrt{94}$ |
| №4 |  | The extension of the side of length 6 | $12,14,2 \sqrt{154}$ |
| №5 |  | The vertex is at the extension of both sides of lengths 4 and 7 | $8,12,14$ |
| №6 |  | The extension of the side of length 4 | $8,12,2 \sqrt{55}$ |
| №7 |  | The extension of the side of length 7 | $12,14,2 \sqrt{154}$ |
| №8 |  | The vertex is at the extension of both sides of lengths 6 and 7 | $8,12,14$ |
| №9 |  | The extension of the side of length 6 | $8,12,2 \sqrt{55}$ |
| №10 |  | The extension of the side of length 7 | $8,14,2 \sqrt{94}$ |
|  | Two points are vertices, one is the midpoint of a side |  |  |
| №11 |  | The vertices are on the side of length 4, the side of length 7 is extended | $4,14, \sqrt{154}$ |
| №12 |  | The side of length 6 is extended | $4,12, \sqrt{154}$ |
| №13 |  | The vertices are on the side of length 6, the side of length 4 is extended | $6,8, \sqrt{94}$ |
| №14 |  | The side of length 7 is extended | $6,14, \sqrt{94}$ |
| №15 |  | The vertices are on the side of length 7, the side of length 4 is extended | $7,8, \sqrt{55}$ |
| №16 |  | The side of length 6 is extended | $7,12, \sqrt{55}$ |
| №17 | All points are midpoints of sides |  | $8,12,14$ |

Thus, there are 17 ways of construction, leading to 11 different triangles.