240. COROLLARY II. To divide a line into any number of Draw AC, making any convenient angle with AB. On AC lay off five equal distances, AD, DE, EF, FG, and GH. Draw HB. Draw GS, FR, EN, and DM parallel to HB. AB is divided into five equal parts. NR (§ 239). In a similar manner prove NR= RS, and RS = SB. Q.E.F. A C F B 244. EXERCISE. The medial lines of a triangle intersect in a common In A HBC, prove OE parallel to HC. AOCH is a parallelogram. .. F is the middle point of AC. Q.E.D. 245. EXERCISE. The point of intersection of the medial lines divides each median into two segments that are to each other as two is to one. 246. EXERCISE. Given the middle points of the sides of a triangle, to construct the triangle. As the variety of exercises in Geometry is practically unlimited, it is impossible to give for their solution any general rules, as may usually be done for problems in Elementary Algebra or Arithmetic. Yet the following hints may be of use to the beginner: 1. Thoroughly digest all the facts of the statement, separating clearly the hypothesis from the conclusion. 2. Draw a diagram expressing all of these facts, including what is to be proved. 3. Draw any auxiliary lines that may seem to be necessary in the proof.1 4. Assuming the conclusion to be true, try to deduce from it simpler relations existing between the parts of the figure, and finally some relation that can be established. (This is the Analysis of the Proposition.) 1 The student should remember in drawing auxiliary lines that a straight line may be drawn fulfilling only two conditions. Two conditions are said to determine a straight line. 5. Then, starting with the relation established, reverse the analysis, tracing it back, step by step, until the conclusion is reached. EXERCISES 1. If two angles of a quadrilateral are supplementary, the other two are also supplementary. 2. Two parallels are cut by a transversal. Prove that the bisectors of two interior angles on the same side are perpendicular to each other. 3. An exterior base angle of an isosceles triangle is 14 R.A.'s. Find the angles of the triangle. 4. If the angles adjacent to one base of a trapezoid are equal, the angles adjacent to the other base are also equal. [§ 122.] 12. From any point D on the base of the isosceles triangle ABC, DE and DF are drawn parallel to the equal sides BC and AB respectively. Prove that the perimeter of DEBF is constant and equals AB + BC. A 13. The angle formed by the bisectors of two consecutive angles of a quadrilateral is equal to one half the sum of the other two angles. [§§ 138 and 157.] 14. How many sides has the polygon the B sum of whose interior angles exceeds the sum of its exterior angles by 12 right angles? 15. On the sides of the square ABCD, the equal distances AE, BF, CG, and DH are laid off. Prove that the quadrilateral EFGH is also a square. E B |