What Is Mathematics? An Elementary Approach to Ideas and Methods, 2nd Edition (1996)

§5. APPLICATIONS

1. Preliminary Remarks

With the introduction of elements at infinity it is no longer necessary to state explicitly the exceptional cases that arise in constructions and theorems when two or more lines are parallel. We need merely remember that when a point is at infinity all the lines through that point are parallel. The distinction between central and parallel projection need no longer be made, since the latter simply means projection from a point at infinity. In Figure 72 the point O or the line PQR may be at infinity (Fig. 85 shows the former case); it is left as an exercise for the reader to formulate in “finite” language the corresponding statements of Desargues’s theorem.

Fig. 85. Desargues’s configuration with center at infinity.

Not only the statement but even the proof of a projective theorem is often made simpler by the use of elements at infinity. The general principle is the following. By the “projective class” of a geometrical figure F we mean the class of all figures into which F may be carried by projective transformations. The projective properties of F will be identical with those of any member of its projective class, since projective properties are by definition invariant under projection. Thus, any projective theorem (one involving only projective properties) that is true of F will be true of any member of the projective class of F, and conversely. Hence, in order to prove any such theorem for F, it suffices to prove it for any other member of the projective class of F. We may often take advantage of this by finding a special member of the projective class of F for which the theorem is simpler to prove than for F itself. For example, any two points A, B of a plane π can be projected to infinity by projecting from a center O onto a plane π′ parallel to the plane of O, A, B; the straight lines through A and those through B will be transformed into two families of parallel lines. In the projective theorems to be proved in this section we shall make such a preliminary transformation.

The following elementary fact about parallel lines will be useful. Let two straight lines, intersecting at a point O, be cut by a pair of lines l₁ and l₂ at points A, B, C, D, as shown in Figure 86. If l₁ and l₂ are parallel then

Fig. 86.

and conversely, if then l₁ and l₂ are parallel. The proof follows from elementary properties of similar triangles, and will be left to the reader.

2. Proof of Desargues’s Theorem in the Plane

We now give the proof that for two triangles ABC and A′B′C′ in a plane situated as shown in Figure 72, where the lines through corresponding vertices meet in a point, the intersections P, Q, R of the corresponding sides lie on a straight line. To do this we first project the figure so that Qand R go to infinity. After the projection, AB will be parallel to A′B′, AC to A′C′, and the figure will appear as shown in Figure 87. As we have pointed out in Article 1 of this section, to prove Desargues’s theorem in general it suffices to prove it for this special type of figure For this purpose we need only show that the intersection of BC and B′C′ also goes to infinity, so that BC is parallel to B′C′; then P, Q, R will indeed be collinear (since they will lie on the line at infinity). Now

Fig. 87. Proof of Desargues’s theorem.

and

Therefore this implies BC||B′C′, which was to be proved.

Note that this proof of Desargues’s theorem makes use of the metric notion of the length of a segment. Thus we have proved a projective theorem by metric means. Moreover, if projective transformations are defined “intrinsically” as plane transformations that preserve cross-ratio (see p.177), then this proof remains entirely in the plane.

Exercise: Prove, in a similar manner, the converse of Desargues’s theorem: If triangles ABC and A′B′C′ have the property that P, Q, R are collinear, then the lines AA′, BB′, CC′, are concurrent.

3. Pascal’s Theorem†

This theorem states: If the vertices of a hexagon lie alternately on a pair of intersecting lines, then the three intersections P, Q, R of the opposite sides of the hexagon are collinear (Fig. 88). (The hexagon may intersect itself. The “opposite” sides can be recognized from the schematic diagram of Fig. 89.)

By performing a preliminary projection we may assume that P and Q are at infinity. Then we need only show that R also is at infinity. The situation is illustrated in Figure 90, where 23 || 56 and 12 || 45. We must show that 16 || 34. We have

Therefore

so that 16 || 34, as was to be proved.

Fig. 88. Pascal’s configuration:

Fig. 89.

Fig. 90. Proof of Pascal’s theorem:

4. Brianchon’s Theorem

This theorem states: If the sides of a hexagon pass alternately through two fixed points P and Q, then the three diagonals joining opposite pairs of vertices of the hexagon are concurrent (see Fig. 91). By a projection we may send to infinity the point P and the point where two of the diagonals, say 14 and 36, intersect. The situation will then appear as in Figure 92. Since 14 || 36 we have a/b = u/v. But x/y = a/b and u/v = r/s. Therefore x/y = r/s and 36 || 25, so that all three of the diagonals are parallel and therefore concurrent. This suffices to prove the theorem in the general case.

Fig. 91. Brianchon’s configuration.

Fig. 92. Proof of Brianchon’s theorem.

5. Remark on Duality

The reader may have noticed the remarkable similarity between the theorems of Pascal (1623–1662) and Brianchon (1785–1864). This similarity becomes particularly striking if we write the theorems side by side:

Not only the theorems of Pascal and Brianchon, but all the theorems of projective geometry occur in pairs, each similar to the other, and, so to speak, identical in structure. This relationship is called duality. In plane geometry point and line are called dual elements. Drawing a line through a point, and marking a point on a line are dual operations. Two figures are dual if one may be obtained from the other by replacing each element and operation by its dual element or operation. Two theorems are dual if one becomes the other when all elements and operations are replaced by their duals. For example, Pascal’s and Brianchon’s theorems are dual, and the dual of Desargues’s theorem is precisely its converse. This phenomenon of duality gives projective geometry a character quite distinct from that of elementary (metric) geometry, in which no such duality exists. (For example, it would be meaningless to speak of the dual of an angle of 37° or of a segment of length 2.) In many textbooks on projective geometry the principle of duality, which states that the dual of any true theorem of projective geometry is likewise a true theorem of projective geometry, is exhibited by placing the dual theorems together with their dual proofs in parallel columns on the page, as we have done above. The basic reason for this duality will be considered in the following section (see also p. 217).