If we take 3 parallel lines and project them by parallel projection or subject them to another affine transformation (for instance if we stretch the (x,y) plane by sending (x,y) to (2x,3y)), then the distances between the lines will change, but any ratio of the distances will not. We can use this to measure relative distance from a point to a line.

Suppose we have 3 parallel lines in the plane: a, b, c. If line m1 is a transversal intersecting these three lines in points A1, B1, C1, we can form the simple (or affine) ratio A1C1/A1B1. If line m2 is another transversal intersecting these three lines in points A2, B2, C2, we can form the simple (or affine) ratio A2C2/A2B2. We know from the theory of parallels (B&B Chapter 4, for instance) that the two ratios are equal.

So there is a basic affine (distance) ratio from 3 parallel lines, which we will denote by ac/ab, that can be computed from a point ratio on any transversal. If the transversal m1 is perpendicular to the parallel lines, the distances A1B1 and A1C1 are actually the distances between the lines, but the ratios are the same for any transversal. We can think of the distance between two parallel lines as being the width of the strip between the two lines. Then the affine ratio ac/ab is the ratio of the widths of two strips, which does not depend on the units of distance measure. In fact this ratio is preserved by affine transformations, since the ratio A1C1/A1B1 is preserved.

Special cases: If two lines coincide, they are not parallel since they meet everywhere. But we can still form this ratio using a transversal.

• If line b = c, then B1=C1, and ac/ab = A1B1/A1/B1 = 1.
• When a = c, then since A1=C1, |A1C1| = 0, so ac/ab = A1A1/A1B1 = 0.
• If a = b but a not = c, then since A1 = B1 and |A1B1| = 0, we can say the ratio is undefined or else say the ratio = |A1C1|/0 = infinity.
• If all 3 lines are the same, we can only say the ratio is undefined.

Suppose we are given a line a and points B and C in the plane. If we want to measure the distance from B or C to a, this will be changed when we project the figure by parallel projection or transform it by an affine transformation. However, the ratio of the two distances will be preserved, and this will be the same as the line ratio discussed in the previous section.

This is how it works. Let b be the line through B parallel to a and c be the line through C parallel to a. Then the ratio of distances from C to line a and B to line a is ac/ab (as defined above) and this ratio is not changed by affine transformations.

We often apply this by starting with a given line a and a given point B as reference landmarks. Then for any point C, the ratio ac/ab measures the distance from m to C, taking the directed distance from a to B as +1.

We can think of this as a function. Let F(C) = ac/ab. Then F(P) = 1 if P is on b. F(P) = 0 is P is on a. If we think of a building with many stories, then if a = ground level and B is at the top of the first story, then F(P) is the height of P (measured in stories) above a.

Any piece of notebook paper is ruled with parallel line segments. Choose 3 lines and label them a, b, c. See that you can compute the ratio AC/AB by simply counting the spaces between the lines. Let's say counting down is positive. As in the figure, if you count down 5 lines from a to b and count up 3 lines from a to c, then the signed ratio is -3/5.

Now draw a transversal and label the intersection points A, B, C. Check that the notebook paper lines, which are evenly spaces, divide the transversal into equal segments. Segment AB = 5 of these segments in one direction of segment AC = 3 of these segments in the opposite direction, so AC/AB = -3/5 also.

Taking another piece of notebook paper, label a line as a again, but this time, choose a point B on one line and a point C on another line. In the figure, B is on the line counted 5 steps below a and C is on the line counted 3 steps above a. So letting the lines parallel to a through B and C be lines b and c, then the ratio ac/ab = -3/5 and this is the ratio of the distances from the two points to line a.

Let A1 and A2 be any two points on line a. Since the triangles A1A2B and A1 A2C have a common base A1A1, the ratio of the area[A1A2C]/area[A1A2B] also equals -3/5. Notice that we count a triangle area if the order of vertices is counterclockwise and the area is negative if the order of vertices is clockwise. So in this case the ratio area[A2A1C]/area[A1A2B] = 3/5.

Special case: If B and C are on the same parallel line, the areas are equal!

Let ABC be a fixed triangle and let D be a point in the plane, for example, let D be a point inside the triangle. Let us call line b = line BC. Let line d be the line through D parallel to b. Also, let line a be the line through A parallel to b.

Denote the intersection of d with line AB as D' and the intersection of d with line AC as D''. Then we can apply the discussion of affine ratios to this situation in several ways.

If we let a the line through A parallel to line BC, then the following ratios are equal:

bd/ba = BD'/BA = BC''/CA = area[BCD]/area[ABC].

This ratio equals 0 if D is on line b and equals 1 if D is on line a.

If D moves along line d, the ratio is not changed. The area of triangle BCD = area[BCD'] = area [BCD''] since these triangles have the same altitude and the same base.

Let's call this ratio m_A(D) and think of it as the relative distance from D to line BC (where the distance is standardized by letting the distance from A to line BC equal 1.)

Here are some examples of the value of m_A(D) for D in various locations.