If a triangle has its midpoints labeled as D, E, and F, how many lines can be constructed connecting these midpoints?

To determine how many lines can be constructed connecting the midpoints D, E, and F of a triangle, we first recognize that each midpoint connects directly to the two other midpoints.

A triangle has three midpoints, one at each side (D for side BC, E for side AC, and F for side AB). Each midpoint can be connected to the other two midpoints, creating a series of segments.

• From D, a line can be drawn to E and another line to F.

• From E, a line can be drawn to D and another line to F.

• From F, a line can be drawn to D and another line to E.

In total, each of the three midpoints connects to two others, leading to a combination of connections:

1. D to E

2. D to F

3. E to D (already counted with D to E)

4. E to F

5. F to D (already counted with D to F)

6. F to E (already counted with E to F)

Counting the unique connections results in 3 lines: DE, DF, and EF. However, if we consider the directions of each line (i.e., D to E and E to