Theorem 5.7.4

Suppose PQ is a segment on line n:

a) If m is parallel to n and D,C are elements of m, then area (CPQ) = area (DPQ).

b) If distinct points D and C lie on the same side of n and area (CPQ) = area (DPQ), then line CD is parallel to n.

To convince yourself drag point B to change the distance between the parallel lines. What happens to the areas of triangles CPQ and DPQ.

Now drag point P to change the distance of line segment PQ. What happens to the areas of triangles CPQ and DPQ?

Finally, drag point C. What happens to the areas of triangles CPQ and DPQ?

Michael Smith, Created with GeoGebra