Figure 29 is called a SIMPLE CLOSED CURVE, “SIMPLE” because it is formed by one continuous line which does not cross itself. It is easy to see that the point marked A is inside the curve.
Figure 30 is also a simple closed curve, but now it is not at all easy to see if point A is inside or outside.
A Frenchman named Jordan discovered an easy way of being sure. Just draw a straight line from the point to the outside curve. If it crosses the curve an odd number of times then the point is INSIDE. If the line crosses the curve an even number of times then the point is OUTSIDE.
a) Figure 31 shows a simple closed curve and two points A and B. Which point is outside the curve? Can you reach B from A without crossing the curve?
b) Figure 32 also shows a simple closed curve and four points A, B, C, D. Which of the four points are inside the curve? How can you tell?
c) Draw other simple closed curves. Try to find a rule for deciding which points are inside your curves.
SUGGESTION: Ask students to draw several different SIMPLE CLOSED CURVES. Choose a point inside each curve and draw a straight line from the chosen point to the outside of the curve. Count how many times the line cuts the curve. Can you now discover a rule?