Julia drew on the line she'd draw to get quarters. Can you see that there are now four identical triangles? (Concentrate on the thickest lines.)

 

 

However, Anna wrote to say:
The least amount of line needed is one line dividing the central two triangles of the square in half.

I knew that the diagram had 18 triangles which is a number not divisible by 4. So I divided two triangles in half to have a total of 20 triangles which is a number divisible by 4.  At first I drew the smaller triangles at the side of the square but this did not work.  Then I divided all of the triangles in half using 9 lines giving me 36 triangles.  I successfully divided the diagram into equally sized quarters of the same shape, but this did not give me the smallest amount of line.  Once more I split two triangles in half but this time the halved triangle was in the middle.