CONGRUENCE OF TRIANGLES.

57

309. Cor. III. If a straight parallel to one side of a triangle cuts off any fractional part of a side, it cuts off the same fraction of the other side.

310. Inverse. The sect joining the mid points of any two sides of a triangle is parallel to the third side, and equal to half of it.

311. Corollary. The sect joining points which bound with any vertex of a triangle the same fractional parts of two sides is parallel to the third side and is that fractional part of it.

ROTATION-CENTER.

312. Theorem. In a plane, the result of sliding any polygon is the same as of a rotation about a fixed point.

Proof. Join vertex A' with its trace A", and B' with B". The perpendicular bisectors of A'A" and B'B' intersect in the rotation-center 0. For ▲ A'OB'AA"OB" [having three sides respectively equal].

=

Consequently X A'OA" B'OB". 313. The altitudes of a triangle are concurrent, and the point is called the triangle's orthocenter.

They must cointersect, since each contains the circumcenter of a triangle made by drawing through the vertices of the given triangle parallels to its sides.

H

B

E A

M

FIG. 153.

1. If, with the vertex of an angle as center, two circles be described,. and the points in which they cut its arms be joined, the joins are || or intersect on the angle's bisector.

2. In a two sides, two altitudes, two medians, two 4-bisectors are, and cross on the axis.

3. Intersecting equal circles are with regard to their common chord.

4. If about two given points as centers pairs of equal intersecting Os be described, all the pairs have their common points on one straight.

5. If of two convex polygons one is wholly within the other, then the outer has the greater perimeter.

6. An interior angle of a regular dodecagon is what fraction of a r't ?

7. If two sides of a ▲ be produced through their common vertex. until each is doubled, the join of the ends is || to the third side.

8. The join of the points of contact of || tangents to a © is a diam

eter.

9. If in a ▲ we decrease one of the equal sides and increase the other equally, the join of the points so obtained is bisected by the third side.

10. In ▲ ABC, if r't bi' of a cuts the st' b in D and c in E, then is

ABD or ¥ ACE = dif' between 4s B and C.

II. If from two p'ts of a st', is to another st’are =, are ||, or the sects from their cross to the p'ts are =.

of a

either the st's

12. The sum of the is to the = sides from any p't in the third side equals one of the = altitudes.

13. All equal sects between two |s belong to two sets of ||s.

14. If from the vertices in the same sense on the sides of a g'm a given sect be taken, the points so obtained are vertices of a g'm cosymcentral with the first.

15. Find the bisector of an ☀ without using its vertex.

16. A quad' with two sides || and the others = is either a g'm or a symtra.

17. If two sides of a quad' have a common r't bi', it is a symtra. 18. The r't bi's of the non-|| sides of a symtra cross on the r't bi' of the other sides.

EXERCISES ON BOOK I.

19. If the diagonals of a quad' are =, its medians are 1.

59

20. If in a trapezoid three sides are =, then the angles adjacent to the fourth side are bisected by the diagonals.

21. The sects to the intersection points of a secant from the 1 projections of ends of a diameter on it are =.

22. A quad' is fixed by 5 given magnitudes. 23. An n-gon is fixed by 2n

3 given magnitudes.

24. The bisector of an of a ▲ and the r't bi' of the opposite side cross on the circum-.

25. The cross of an altitude (produced through its foot) with the circum- is to the orthocenter with respect to that side of the ▲.

26. Whether their vertex be on or within the O, a pair of vertical angles together intercept the same part of a .

27. Vertical r't s with vertex on or within a C intercept half of it. 28. Joining one common p't of two = intersecting Os to the crosses of a secant through the other common point gives = sects.

29. If from one intersection p't of two = Os as center we describe any third circle cutting them, then the four intersection p'ts are vertices of a symtra whose non- sides go through the other intersection p't of the = OS.

30. A on the common chord of two Os as diameter bisects all sects through an intersection p't of the Os and ending in them.

31. A symtra is cyclic.

32. A deltoid is a circumscribed quad'.

33. The four X-bisectors of a quad' make a cyclic quad'.

34. The four crosses of the inner with the outer common tangents to two Os lie on a circle with their center-sect as diameter.

35. The sect of an outer between the inner tangents equals the sect of an inner between its points of contact.

36. Each side of a ▲ is, by the p'ts of contact of the in- and an ex-O, divided into three sects, of which the outer two are =.

37. If a polygon has a circum

and a concentric in- O, it is regular.

38. To make a regular hexagon, trisect the sides of a regular trigon and join the points next its vertices.

39. To make a regular octagon, about each vertex of a square, with half the diagonal as radius, describe a O and join the crosses next its vertices.

40. If a p't of its circum- be joined to the vertices of a regular ▲, the greatest sect equals the sum of the other two.

BOOK II.

PURE SPHERICS.

CHAPTER I.

PRIMARY CONCEPTS.

314. A circle is a closed line that will slide in its trace. Though in itself unbounded and everywhere alike, yet it is finite. On it two points starting from coincidence and moving in opposite senses will meet.

315. Every point on a circle has one other on it such that the two bisect the circle. Two such are called opposite points on the circle.

316. If a pair of opposite points can be kept fixed while a circle moves, it describes a surface called a sphere.

317. A sphere is a closed surface which will slide in its trace. Though in itself unbounded and everywhere alike, yet it is finite, being generated completely by one finite motion of a finite line.

318. ASSUMPTION I. Any figure drawn on the sphere may be moved about in the sphere without any other change.

319. Assumed Construction I. Through any two points, in a sphere, can be passed a line congruent with the generating line of the sphere.

In Book II. g-line will always mean such a line, and sect will mean a piece of it less than half.

PRIMARY CONCEPTS.

61

320. ASSUMPTION II. Two sects cannot meet twice on the sphere.

If two sects have two points in common, their g-lines coincide throughout. Through two points, not opposite points of a g-line, only one distinct g-line can pass.

321. A piece of the sphere with part of a g-line as one of its boundaries, would fit all along the g-line from either side.

322. Because of the symmetry in its generation, the sphereis cut by any g-line on it into two equal parts, called hemispheres.

323. If one end point of a sect be kept. fixed, the other end point moving in the sphere describes what is called an arc, and the sect is said to rotate in the sphere about the fixed end point. The arc is greater as the amount of rotation is greater.

324. Two sects from the same point, when looked at with special reference to the amount and sense of rotation to bring their g-lines into coincidence, are said to form a spherical angle. The spherical angle is greater as the amount of rotation is greater.

FIG. 154.

FIG. 155.

325. When a sect has rotated just sufficiently to fall again into the same g-line, the angle described is called a straight angle, and the arc described is called a semicircle.

326. Half a straight angle is called a right angle.

327. The whole angle about a point in the sphere, that is, the angle described by a sect rotating until it coincides with its trace, is called a perigon; the whole arc is called a circle.