Okay, I believe that it:

is possible

First:

Pretend all the cities are around in a circle, equidistant from their neighbours

Then we will:

Create 4 sets of 25 triangles with all 12 edge lengths, so that we can rotate each triangle around into the 25 possible positions for one particular orientation

This will:

Make a set of 100 triangles which covers each road exactly once

Namely, the sets are:

(a/b represents the arc lengths counting either way around the circle)
1/24, 3/22, 4/21
2/23, 8/17, 10/15
5/20, 9/16, 11/14
6/19, 7/18, 12/13

Demonstration:

Okay, I believe that it:

is possible

First:

Pretend all the cities are around in a circle, equidistant from their neighbours

Then we will:

Create 4 sets of 25 triangles with all 12 edge lengths, so that we can rotate each triangle around into the 25 possible positions for one particular orientation

This will:

Make a set of 100 triangles which covers each road exactly once

Namely, the sets are:

(a/b represents the arc lengths counting either way around the circle)
1/24, 3/22, 4/21
2/23, 8/17, 10/15
5/20, 9/16, 11/14
6/19, 7/18, 12/13

Demonstration:

In the below image, the shorter arc length s is selected for the label d_s. One of the 25 triangles generated from each line is shown, in the order orange, blue, green, red.

Okay, I believe that it:

is possible

First:

Pretend all the cities are around in a circle, equidistant from their neighbours

Then we will:

Create 4 sets of 25 triangles with all 12 edge lengths, so that we can rotate each triangle around into the 25 possible positions for one particular orientation

This will:

Make a set of 100 triangles which covers each road exactly once

Namely, the sets are:

(a/b represents the arc lengths counting either way around the circle)
1/24, 3/21, 4/20
2/23, 8/17, 10/15
5/20, 9/16, 11/14
6/19, 7/18, 12/13

Okay, I believe that it:

is possible

First:

Pretend all the cities are around in a circle, equidistant from their neighbours

Then we will:

Create 4 sets of 25 triangles with all 12 edge lengths, so that we can rotate each triangle around into the 25 possible positions for one particular orientation

This will:

Make a set of 100 triangles which covers each road exactly once

Namely, the sets are:

(a/b represents the arc lengths counting either way around the circle)
1/24, 3/22, 4/21
2/23, 8/17, 10/15
5/20, 9/16, 11/14
6/19, 7/18, 12/13

Demonstration: