LOCUS PROBLEMS 117 257. PROBLEM 35. To find a point such that its joins to three given points not collinear are equal. Let A, B, C be the three given points. B It is required to find a point 0, such that 04, OB, OC are all equal. Draw PP' and QQ' the perpendicular bisectors of the lines AB and BC. The lines PP' and QQ' intersect, for if they were parallel, then the lines AB and BC, being respectively perpendicular to them, would be in one straight line. Let the lines PP' and QQ' intersect in 0. To prove that the point 0, and no other point, satisfies the given conditions. The line PP' contains all those points and only those, whose joins to A and B are equal (253). The line QQ contains all those points, and only those, whose joins to B and C are equal. Therefore the point common to PP' and QQ', and no other point, has its joins to A, B, and C equal. NOTE. This construction is used later (III. 75) in finding the center of a given circle. Ex. 1. In a given line find a point whose joins to two given points are equal. Ex. 2. Find a point from which the perpendiculars to two given 258. PROBLEM 36. To find a point from which th perpendiculars to three given lines (forming a tr angle) shall be equal. Let the three lines LL', MM', NN' form a triangle ABC. It is required to find a point 0, such that the perpendicu lars from 0 to these lines are equal. [The construction and proof are left to the student. Show tha there are four solutions.] Ex. 1. If two of the three lines are parallel, how many solutions are there? Ex. 2. On a given line find those points from which the perpendiculars to another given line are equal to an assigned line-segment. Ex. 3. On a given line how many points are there from which the perpendiculars to two given lines are equal ? THEOREMS ON CONCURRENCE 259. Definition. Three or more lines that meet in a common point (when prolonged if necessary) are said to be concurrent. The principle of the intersection of loci may be used to prove the first two of the following theorems relating to the concurrence of certain lines in a triangle. The third is Concurrence of perpendicular bisectors of sides. 260. THEOREM 50. In any triangle the three perpendicular bisectors of the sides are concurrent. Outline. Use the construction of 257. Then show that O lies on the perpendicular bisector of the side AC. Concurrence of angle-bisectors. 261. THEOREM 51. In any triangle the three bisectors of the interior angles are concurrent. Outline. Let the point O in 258 be the intersection of two of the angle-bisectors. Show that O lies on the third angle-bisector. 262. Cor. The exterior angle-bisectors through two vertices and the interior angle-bisector through the third vertex are con current. 263. Definition. The lines drawn from the vertices of a triangle perpendicular to the opposite sides, respectively, are called the principal perpendiculars of the triangle. Concurrence of principal perpendiculars. 264. THEOREM 52. The three principal perpendiculars of a triangle are concurrent. Let ABC be any triangle. Let AD, BE, and CF be the principal perpendiculars. F E To prove that AD, BE, and CF are concurrent. MOM. ELEM. GEOM.-9 Through the vertices A, B, C draw lines parallel respectively to the opposite sides, so as to form a second triangle A'B'C'. Outline. Prove by 153 that A is the mid-point of B'C', B the midpoint of A'C", and C of A'B' ; hence that AD, BE, CF are the perpendicular bisectors of the sides of the new triangle A'B'C'. Then draw desired conclusion and quote authority. ON METHODS OF ANALYSIS 265. When a new theorem or problem is presented for investigation (as in the miscellaneous exercises that follow), we try to discover some connection or relationship between the new proposition and the previous ones with which we are familiar. This relationship is to be discovered by means of a preliminary analysis. The words analysis and synthesis and the corresponding adjectives analytic and synthetic are much used in mathematics. In general, analysis means the sepa ration of a whole into its parts, and synthesis means bringing the parts together to make a whole. In geometry the words are used in a more restricted sense. In synthesis we begin with admitted facts, and, by the aid of principles or theorems already accepted and problems already solved, we prove some new theorem or solve some new problem. This is usually the most convenient way of presenting the result when it is once obtained; but the actual discovery is often made in the reverse way by means of an analysis, in which we begin with the conclusion and then examine the different conditions that are necessary or sufficient to lead to the result in question. The analysis of a problem was described in 131, and illustrated in various subsequent articles. The analysis of a theorem is somewhat similar, and may be conducted in two ways, which may be called, respectively, the analysis of antecedents and the analysis of consequents. 266. Analysis of antecedents. In this method we examine the antecedent conditions from which the conclusion in question would follow, and then compare these conditions with the given hypothesis. For example, let the conclusion be called 'statement S,' then the analysis of antecedents may be put in the following form : The statement S is true, if the statement R is true; and so on. If by this method we get back to some antecedent statement A which we know to be true by some principle already accepted, or which would follow from the given hypothesis, then we are warranted in asserting the truth of statement S. The successive steps from A to S can then be presented in the reverse of the order just given, and the proof can be arranged in the usual synthetic form beginning with the hypothesis and ending with the conclusion to be demonstrated. If, however, we come only to a statement that we know to be false (or do not know to be true), then the statement S may or may not be true, and nothing is proved. A new set of antecedent conditions may then be examined. This method often proves the truth of a theorem; it cannot by itself prove any statement false. 267. Analysis of consequents. In this method we examine the consequences that would follow if the theorem were supposed to be true, and then compare these consequences with the hypothesis and other accepted facts. The analysis of consequents may be put in the following form: If the statement S is true, then the statement T is true; and so on. If by this method we arrive at some statement that we know to be false (or inconsistent with the hypothesis), then we conclude that the statement S is false, since it can be reduced to an absurdity. If, however, we come only to a statement Z that we know to be true (or do not know to be false), then the statement S may or may not be true, and nothing is proved. This method of analysis often proves the falsity of a statement; it cannot by itself prove any statement true, since the steps taken from S to Z are not always reversible; it sometimes, however, points the way to a synthetic proof by reversal of the steps. 268. Analysis of the opposite. Either of the two methods of analysis may be applied to the opposite of the statement S. The analysis of antecedents gives a decisive result if we arrive at an antecedent known to be true, for then the opposite of S is true, and S is |