Configuration Theorems (Topics in Mathematics) by B. I Argunov

By B. I Argunov

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Corresponding sides of these triangles intersect at the points P, Q, and R. According to Desargues's theorem, these points are situated on one line, which was to be proved. connecting the ends of a base of a trapezoid or a parallelogram with the midpoint of the opposite base intersect the lateral sides at points on a line which is parallel to the base. Indeed, if ABCD is a trapezoid with bases AD and BC (Fig. 31), then, as is known, the point M, constructed as in TheoCOROLLARY. l On the other band, the point of intersection R of the lines AD and BC is in this case infinitely remote, so that the line PQ, which must pass through this point, will be parallel to the lines AD andBC.

Then the product ab of the given numbers can also be represented as a line segment. This segment can be 37 constructed as follows: Take an arbitrary angle MON. On side OM layoff the first factor of ab, that is, the segment OA = a, and on the other side, ON, the secondfactor, OB = b, and the unit, OE = I. Construct the line BB, II AE. Then OB, OA (jjf = OE OB, a. or -b- = T' hence, OB, = abo By the same method, construct the product ba. For this layoff on side OM the first factor of ba, that is, the segmeut OB' = b, and on the side ON the second factor, OA' = a.

Thus, "in algebraic language" the Pappus-Pascal theorem becomes the commutative law of multiplication. = 21. = SCHEMATIC NOTATION FOR CONFIGURATION THEOREMS After deeper study of the problem and upon acquiring the necessary experience, a person can learnto statevariousalgebraic 38 identities as configuration theorems. Since one can construct as many identities as one likes, one can also construct as many con- figuration theorems as one wants. Furthermore, the number of points and lines involved in a theorem can be made arbitrarily large.

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