Above is the sample I'll be working with. I want to divide the paths wherever they overlap each other.

Normally, you could just use the Knife tool and just cut them by eye. Sometimes this isn't quite as accurate as you'd want it.

We'll still be using the Knife tool, but you'll use another feature in FreeHand to help make the cuts more accurate, Smart Cursors.

Before you try this, go under the Preferences -> General and turn on the Smart Cursor option.

First select one of the paths and Clone it. While it's still selected, click on the Guide layer. When you hide the other layers, you'll see that we've turned that clone into a guide.

Now go under the View menu and turn on the option Snap to Guides.

Double-click on the Knife tool and turn off the option Close cut paths.

Now select the other path that you didn't turn into a guide, above I selected the Triangle. Move your cursor at an intersection. When you see the pointer change to the one shown above (with the little o), that means it's snapping to the Guide.

Just click once and it will cut the path. Continue cutting until you are finished with that path.

For the other path, it's a little different. Since we used that one for the guide, you can't really get accurate results because you can "Snap to" the guide at any point of the ellipse. So instead we use the cut points of the Triangle as our "Snap to" points.

Under the View menu, turn on Snap to point. Now, when you move the cursor over the intersections, you should see the cursor change (with the little dot). Cut when you see it change.

When you are finished, you will have separate line segments wherever the two paths intersected like shown above.

It's now safe to delete that guide once you are done.