By Melvin Hausner

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12, find the ratios AG/GE and AF/FC. We should like A and B to have mass 2 and mass 3, respectively. Clearly, any proportional numbers can be used. Since B and C need mass 5 and mass 2, we see that B “wants” to have mass 5 as well as 3. Choose 15, the least common multiple for the mass of B (in order to avoid fractions), and we quickly obtain Fig. 13, where we read In the exercises which follow, try to work as “physically” as possible, where this can be done. 13 Exercises 1. In Fig. 14, AP = 2PB and QC = 2PQ.

Since nB and pC are both solutions, we have xX = nB = pC This completes the proof. It is clear from this proof that this theorem is simply a restatement of the uniqueness part of Axiom VI. ” Exercises In Exercises 1 to 5 below, express each of the geometric statements in algebraic language. No proofs are required. Many of your answers will be somewhat clumsy, but this will be improved in Chapters 2 and 3. 1. P is inside the tetrahedron QRST. 2. Segments and intersect. 3. , all on a line); D, E, and F are not collinear.

The operation of “+” is familiar to anyone who has ever mixed 2 gallons of brick-red paint with 5 gallons of flaming orange. The result is 7 gallons of a new color. With the exception of axiom VI (subtraction), the axioms are intuitively seen to be true: I. It is clear that m gallons of P and n gallons of Q determine m + n gallons of some unique color R (the closure law). II, III. It does not matter which color is poured into which and, if three containers of paint are at hand, the final color is determined uniquely regardless of which two are mixed first (the commutative and associative laws).