Journey into Mathematics by Joseph J. Rotman

Journey into Mathematics by Joseph J. Rotman

Author:Joseph J. Rotman [Rotman, Joseph J.]
Language: eng
Format: epub
ISBN: 9780486151687
Publisher: Dover Publications
Published: 2012-09-25T16:00:00+00:00


Figure 3.16

This inequality holds, for

∣AC∣ + ∣CB∣ = ∣AC∣ + ∣CE∣ + ∣EB∣ > ∣AE∣ + ∣EB∣

(the shortest path from A to E is the straight line AE). But

∣AE∣ + ∣EB∣ = AD∣ + ∣DE∣ + ∣EB∣ > ∣AD∣ + ∣DB∣

(the shortest path from D to B is the straight line DB). We have verified the inequality ∣AC∣ + CB∣ > ∣AD∣ + ∣DB∣.

Now let β be a 3-edged concave path, as in Figure 3.17.

Figure 3.17

Extend AD, and let it meet CB in the point F. Because the 3-edged path β is concave, it is entirely inside the triangle ∆AFB. Let us compute.



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