153. COR. Each exterior angle of an equiangular polygon Ex. 218. How many sides has a polygon, the sum of whose interior angles equals twice the sum of its exterior angles? Ex. 219. How many sides has an equiangular polygon, three of whose exterior angles are together equal to 90° ? Ex. 220. How many sides has an equiangular polygon, four of whose angles are together equal to seven right angles? 154. The bisectors of the angles of a triangle meet in a common point, which is equidistant from the sides of the triangle. A To prove 1°. Point O is a common point of BD, CE, and AF; and 2o. Point O is equidistant from BC, AB, and AC. Proof. Let BD and AF meet in O. ... Every point in the bisector of an angle is equidistant from the sides of the angle. .. O is a common point of BD, CE, and AF; and is equidistant from BC, AB, and AC. Q.E.D. Ex. 221. If in the diagram for Prop. XLII (1) ≤ A = 60°; ≤ B = 50°; (2) ≤ A = m°; ≤B = n°. Find all other angles of the figure. PROPOSITION XLIII. THEOREM 155. The perpendiculars erected at the midpoints of the sides of a triangle meet in a common point which is equidistant from the vertices of the triangle. DG, EH, and FI are perpendicular-bisectors of AB, BC, and CA, respectively. To prove 1°. O is common to DG, EH, and FI; and 2o. O is equidistant from A, B, and C. Proof. also Let DG and EH intersect at 0. (119) (119) (112) .. O is common to DG, EH, and FI; and is equidistant from A, B, and C. Q.E.D. Ex. 222. Construct a circumference passing through the vertices of a given triangle. Ex. 223. In what kind of a triangle will the point of intersection of the perpendicular-bisectors be within the triangle? without? in one side? ANALYSIS OF THEOREMS 156. An analysis of a theorem is the course of reasoning by which a proof is discovered. We inquire what means we have to prove a conclusion, as illustrated in the following example. 157. THEOREM. The bisectors of the opposite angles of a parallelogram are parallel. Hyp. The angles B and D of the parallelogram ABCD are bisected. To prove ED || BF. A E B F Analysis. 1. The means for proving that two lines are parallel are: (a) A parallelogram. (b) Equal alternate interior angles. (e) Corresponding angles.. (d) Supplementary interior angles, etc. Let us select any of these methods, e.g. 1 (a), i.e. to prove EBFD is a parallelogram. 2. The means for proving a quadrilateral is a parallelogram are: (a) Opposite sides are equal. 3. The means for proving the equality of lines is usual pair of equal triangles, i.e. 4. The equality of the two triangles is easily established The above demonstration is not the only one nor the sho Each of the possibilities indicated under 1 (a), 1 (b), one. 1(d), 2 (a), 2 (b), will furnish one or more proofs. Ex. 224. Prove the above proposition by means of 2 (b). 158. THEOREM. If a median is intersected by another one, the segment of the median between the point of intersection and vertex is twice its other segment. Analysis. 1. The means of proving that one line equals twice another is: (a) Bisect the greater (b) Double the smaller (143) Each method furnishes a proof. Select 1 (a). Bisect AO and CO in F and G, respectively, and prove FO= OE, GO = DO. 2. The means of proving the equality of lines is usually a pair of equal triangles, i.e. The equality of the triangles may now be easily established. Hence, |