Suppose the points where we break the line are x and y, where 0 < x < y < 1. So our "possible" region is the following triangle on the plane:

The three sides have lengths x, y-x, and 1-y. We must satisfy the Triangle Inequality on all three sides, meaning that the sum of any two of these lengths must be greater than the third. This gives us

1. : This gives y>1-y, so y> 1/2.

2. : This gives 1-x> x, so x<1/2.

3. : This gives , so .

Adding these lines to the diagram gives us the darker "success" region.

The darker region makes up 1/4 of the total region, so the probability is 1/4.