Solution Exercise 4.22.
Definition of Term Commonly Used in Mathematics:
Clearly – I do not want to write down all the steps.

Exercise 4.22. Write the proof for Theorem 4.5, the existence of a harmonic set of points that was briefly described in this section.

Theorem 4.5. There exists a harmonic set of points.

Proof. By Exercise 4.5, there exists a complete quadrangle EFGH. By the definition of a complete quadrangle, A = EF · GH and B = EH · FG are diagonal points. Also, EG and FH are opposite sides of quadrangle EFGH. Let C = AB · EG and D = AB · FH. Then, {A, B, C, D} is a harmonic set with H(AB, CD) provided the points A, B, C, and D are four distinct points.
        We need to show the four points A, B, C, and D are distinct. We show one case, the other cases are similar. Suppose A = C. Since A = EF · GH and C = AB · EG, A is on line EF and A = C is on line EG. Hence, the set of points {A, E, F, G} is a collinear set, which contradicts that {E, F, G} is a noncollinear set since EFGH is a complete quadrangle. Hence, A and C are distinct points. The other cases are similar.
        Therefore, there exists a harmonic set of points.//