EXERCISES ON BOOK II. 1. Explemental s at the q-poles of = Os intercept explemental 2. Explemental arcs of equal circles have equal spherical chords. 3. As a spherical chord increases its major arc decreases. 4. If Os pass through 2 given p'ts their centers all lie on the r't bi' of the join of the 2 p'ts. 5. If 2 Os touch internally, a to the diameter through the p't of contact has equal pieces between the 2 Os. 6. The g-lines on which is from a fixed p't are equal envelop a O with this p't for center. 7. The centers of Os touching two given g-lines all lie on the bisectors of the s made by these g-lines. 8. The centers of Os touching 3 given g-lines lie on the bisectors of the s made by these g-lines. 9. If a quad' is cyclic, the r't bi's of its sides and of its two diagonals are concurrent. 10. ABCD is a cyclic quad'; AD, BC meet in F. Where does tan at F to circum-0 CDF meet AB? EXERCISES ON BOOK II. 83 11. One convex polygon wholly contained within another has the lesser perimeter. 12. The perimeter of any is less than a g-line. The perimeter of any convex spherical polygon is less than a g-line. 13. If 2 Os touch, and through the p't of contact a g-line be drawn to cut the Os again, where will the tangents at these crosses meet? 14. If 2 Os touch, and through the p't of contact 2 g-lines be drawn cutting the Os again, where will the joins of these crosses meet? 15. If the common chord of 2 intersecting os be produced to any p't, the tangents to the 2 os from this p't are =; and inversely. 16. If the common chord of 2 intersecting os be produced to cut a common tangent, it bisects it. 17. The 3 common chords of 3 os which intersect each other are concurrent. 18. How do the in-, circum-, and ex-radii of a regular compare in size? 19. If a quad' can have a circle inscribed in it, the sums of the oppo site sides are equal. 20. If two equal os intersect, each contains the orthocenters of As inscribed in the other on the common chord as base. 21. Three equal os intersect at a p't H, their other points of intersection being A, B, C. Show that H is orthocenter of ▲ ABC; and that the triangle formed by joining the centers of the circles is to ▲ ABC. 22. The feet of 1s from A of ▲ ABC on the external and internal bi's of s B and C are co-st' with the mid p'ts of b and c. Does this hold for the sphere? 23. If two opposite sides of a quad' are =, they make = s with the median of the other sides. Prove for the plane, then extend to the sphere. 24. (Bordage.) The centroids of the 4 As determined by 4 concyclic p'ts are concyclic. 25. The orthocenters of the 4 As determined by 4 concyclic p'ts A, B, C, D are the vertices of a quad' to ABCD. The in-centers are vertices of an equiangular quad'. 26. (Brahmegupta.) If the diagonals of a cyclic quad' are 1, the from their cross on one side bisects the opposite side. 27. If the diagonals of a cyclic quad' are 1, the feet of the Ls from their cross on the sides and the mid p'ts of the sides are concyclic.` 28. If tangents be drawn at the ends of any two diameters, what sort of a quad' is circumscribed ? 29. In any equiangular polygon inscribed in a O, each side is equal to the next but one to it. Hence, if an equiangular polygon inscribed in a have an odd number of sides it must be equilateral. Any equilateral polygon inscribed in a C is equiangular. 30. In any equilateral polygon circumscribed about a O, each is to the next but one to it. Hence, if an equilateral polygon circumscribed about a circle have an odd number of sides, it must be equiangular. Any equiangular polygon described about a C is equilateral. 31. The circle through any 3 vertices of a regular polygon contains the remaining vertices. 32. If one of 2 equal chords of a bisects the other, then each bisects the other. 33. Given 2 symcentral g-lines and their symcenter. Find the g-line symcentral to a third given g-line with respect to this symcenter. ΞΔ 34. All =λ on the same side of the same base have their sides bisected by the same g-line. 35. The tri-rectangular A is its own polar. BOOK III. EQUIVALENCE. 429. Magnitudes are equivalent which can be cut into parts congruent in pairs. 430. Problem. To describe a rectangle, given two consecutive sides. Construction. Draw a straight, erect to it a perpendicular. From the vertex of the right angle lay off one given sect on the straight, the other on the perpendicular. Through their second end points draw parallels, one to the straight, one to the perpendicular. 431. Corollary. A rectangle is completely determined by two consecutive sides; so if two sects, a and b, are given, we may speak of the rectangle of a and b, or we may call it the rectangle ab. Thus, when a and b are actual sects, we mean by ab a definite plane figure with four right angles, four sides, and an enclosed surface. FIG. 199. α a 432. The sum of two polygons is any polygon equivalent to them. 433. Theorem. In any right-angled triangle, the square on the hypothenuse is equivalent to the sum of the squares on the other two sides. Hypothesis. ▲ ABC, r't angled at B. sq' on AC. Proof. By 430, on hypothenuse AC, on the side toward the ▲ ABC, describe the sq' ADFC. C G' D B' A On the greater of the other two sides, as BC, lay off CG= AB. Join FG. Then, by construction, CA FC, and AB = CG, and CAB = 4 FCG, since each is the complement of ACB; . A ABC A CGF. Rotate the ABC about A through a minus r't ; this brings B to B'. Likewise rotate CGF about F through a + r't ; this brings G to G'. The sum of the angles at D = st' X. .. G'D and DB' are in one straight. Produce GB to meet this straight at H; then BC= GF= FG'; and r't GGFG' FGH; . GFG'H equals × :. square on BC. = -= Again, BA AB', and r't B' B'AB ABH; ¥ × = .. ABHB is the sq' on AB. .. sq' of AC = sq' of AB+ sq' of BC. 434. An altitude of a parallelogram is a perpendicular from a point in one side to the straight of the opposite side, which is then called the base. A G 435. Theorem. A parallelogram is equivalent to the rectangle of either altitude and its base. DF B F D A B FIG. 202. C Proof. If CD, the side of the g'm opposite the base AB, contains F, a vertex of the rectangle, then ABFD ≈ ABFD, and ▲ BCF ~ ▲ ADG. If the sides AD, BF intersect in H, then, c by continued bisection, cut BF into equal parts each less than BH. Through these points draw straights to the base, so dividing the rectangle into congruent rectangles, each as above, equivalent to the corresponding parallelogram. 436. Corollary. All parallelograms having equal altitudes and equal bases are equivalent. |