Let T be a triangle and let A be the area of T. Given a point p inside the triangle and a number r that satisfies 0 < r ≤ ½, we seek to count the number N (p, r) of straight lines that pass through p and cut T into two pieces so that one piece has area rA. As the point p and the ratio r vary, the value of N (p, r) ranges from 0 to 6. Given a ratio r, we want to determine the regions within T for which the function N (p, r) assumes various integer values. For example, a region inside T for which N (p, r) = 6 only exists for values of r between and ½, and we would like to quantify the area of this region (as a fraction of the total area of the triangle) as r varies.