PROPOSITION XLI. THEOREM. 147. An intercept parallel to the base of a triangle and bisecting one side bisects the other also. Given: In triangle ABC, an intercept OE parallel to AB and bisecting BC in 0; Complete the parallelogram AD by drawing BD, CD, parallel to AC, AB, respectively, and produce EO to meet BD in F. Since EF is an intercept through the mid point of BC, (Hyp.) Q.E.D. (Ax. 1) (since AF is a parallelogram by construction,) .. AE EC. 148. COR. Conversely, if OE bisects both AC and BC, then OE is parallel to AB, and OE is equal to AB. For OE must coincide with the parallel to AB through 0, since that parallel must pass through E, the mid point of BC (147). Also OE is equal to FE (145), and FE is equal to AB (136). EXERCISE 86. In the diagram for Prop. XLI., show that OEAB can be superposed on OFDC. 87. In the diagram for Prop. XLII., if BD, CG, be drawn, show that these lines will intersect in the mid point of FH. 88. In the same diagram, if AD = BC, then the angles A and B are equal. PROPOSITION XLII. THEorem. 149. An intercept parallel to the bases of a trapezoid and bisecting one of the nonparallel sides bisects the other also. Given: In trapezoid AC, EF parallel to AB, bisecting AD in E and meeting BC in F; Draw DG to BC, meeting AB, EF, in G, H, resp. Since in ▲ DAG, EH is to AG and bisects AD, (Hyp.) 150. COR. Conversely, if EF bisects both AD and BC, then EF is parallel to AB and equal to 1⁄2 (AB + CD). For EF must coincide with the parallel to AB through E, since that parallel must pass through F, the mid point of BC (149). Also EF EH + HF AG + 1⁄2 (BG + CD) = {(AB + CD). = EXERCISE 89. In the diagram for Prop. XLII., if CK be drawn parallel to AD, then will BK be equal to AG, and GK = 2 DC – AB. 90. If through P, the mid point of AE, PQ be drawn parallel to AB to meet BF in Q, then will BQ be one fourth of BC, and PQ will be equal to (3 AB + CD). 151. If a series of parallels make equal intercepts on one transversal, they make equal intercepts on any transversal. Given Transversals AE, ae, cut by a series of parallels Aa, Bb, Cc, Dd, Ee, so that the intercepts AB, BC, CD, DE, on AE are equal; To Prove: The intercepts ab, bc, cd, de, on ae, are also equal. 152. COR. Conversely, if Aa, Bb, etc., make equal intercepts on AE and also on ae, then Aa is || to Bb, Cc, etc. (150) EXERCISES. QUESTIONS. 91. Would a triangle constructed of rods hinged at their extremities be rigid; that is, incapable of change of form? Would a parallelogram similarly constructed be rigid ? 92. Triangles having their sides severally equal have their angles also severally equal. Is the converse true? 93. An acute angle being given, by what construction can you find its complement? Its supplement ? 94. By what angle is the supplement of an angle greater than its complement? 95. A right angle being 90°, how many degrees are there in the complement of an angle of 36° ? Of 45° ? Of 90° ? How many in their supplements? 96. Bisect an obtuse angle (81); also a straight angle. Can you bisect a reflex angle, i.e., an angle greater than a straight angle, by the same process? 97. In Proposition XXIII., can you prove Cor. 2 independently? 98. Are all straight lines that cannot meet parallel? 99. If one angle of a triangle is 230, what is the sum of the other angles? 100. If one angle of a triangle is equal to the sum of the other two, how many degrees has that angle? What is it called? 101. If the vertical angle of an isosceles triangle is 50°, how many degrees in each of the base angles ? 102. If a base angle of an isosceles triangle is 45°, what is the vertical angle? 103. In an isosceles triangle, if each base angle is (1) twice, (2) three times, (3) n times, the vertical angle, how many degrees in each ? 104. How many degrees in each angle of an equilateral triangle ? 105. An acute angle of a right triangle is of the other acute angle. How many degrees in each? Generalize by putting for. m 106. Arrange the following terms in order of generality: square, polygon, rectangle, quadrilateral, parallelogram. 107. I wish to cut off half a rectangular field by a straight fence. Through what point must the fence pass? 108. The base of a triangle is fifty feet. How long is the line joining the mid points of the two sides? 109. How many degrees in the sum of the interior angles of a polygon of four sides? Of five? Of six ? Of n sides? 110. How many degrees in each interior angle of an equiangular polygon of four sides? Of five? Of six? Of ten sides? 111. One angle of a parallelogram is double the other. degrees in each? How many, if one is of the other? m n How many GEOMETRICAL SYNTHESIS AND ANALYSIS. In the demonstration of propositions we have usually proceeded by the method of synthesis, or direct proof; that is, taking as a basis certain admitted truths, we built upon these a demonstration of the truth we wished to establish. In other words, in synthesis we reason from admitted principles to consequences (see, for example, 72). This, however, though usually the most convenient way of presenting the proof of a proposition, does not show how that proof was invented; we are given a result, not the process by which the result was reached. In the method of analysis we proceed in the opposite way; that is, assuming as true the conclusion we wish to establish, we reason back to principles. If we are led back to principles already known as true, we can take these as the basis of a synthetic proof of the conclusion we wish to establish. If, on the other hand, we are led to a contradiction of a known truth, we know that the assumed conclusion was false. This indirect method is often made use of in the demonstration of theorems for which it would be inconvenient or difficult to find direct proof. In Prop. XXVII., for example, we show that CD must coincide with CE because their noncoincidence would entail a consequence that had been shown to be impossible. Instead of proving directly the conclusion we wish to establish, we prove it indirectly by showing that any other conclusion would lead to a contradiction of some known truth. Among the exercises about to be given, most are so easy that the principles on which to base a synthetic proof at once suggest themselves. The student should, however, in every case, base his synthetic proof upon a previous analysis, guided by the general directions given below. The diagrams and references given as aids should, as far as possible, be left as a last resort. |