23. On the sides of a square ABCD points G, H, E, F, P, etc., are taken, so that GA AH : = EB =BF PC, etc. On the diagonals AC and BD points K, L, M, N, are laid off so that OK = OL = OM = ON.

=

(a) Prove that GAHN
(b) Prove that HEKON

EBFK, etc.
FPLOK.

SUGGESTION. Superpose one figure on the

other.

24. The bisectors of the angles of a rhomboid form a rectangle; those of a rectangle form a square.

25. In the figure ABCDEF is a regular hexagon. ABHG, BCLK, etc., are squares.

(a) What kind of triangles are BHK, CLM, etc.? Prove.

(b) Is the dodecagon (twelve-sided polygon) HKLMN... regular? Prove.

(c) How many axes of symmetry has this dodecagon?

(d) Has it a center of symmetry?

(e) Are the points S, F, B, K collinear? (That is, do they lie in the same straight line?)

Parquet Flooring.

M

R

E

S

26. If from any two points P and Q in the base of an isosceles triangle parallels to the other sides are drawn, two parallelograms are formed whose perimeters are equal.

Tile Pattern.

27. The middle point of the hypotenuse of a right triangle is equidistant from the three vertices. (This is a very important theorem.)

28. State and prove the converse of the theorem in Ex. 27.

PQ

29. The bisectors of the four interior angles formed by a transversal cutting two parallel lines form a rectangle.

30. The sum of the perpendiculars from a point in the base of an isosceles triangle to the sides is equal to the altitude from the vertex of either base angle on the side opposite.

31. Find the locus of the middle points of all segments joining the center of a parallelogram to points on the sides. (See § 159, Ex. 8.)

32. In the parallelogram ABCD points E and F are the middle points of AB and CD respectively. Show that AF and CE divide BD into three equal segments.

33. The sum of the perpendiculars to the sides of an equilateral triangle from a point P within is the same for all such points P (i.e., the sum is a constant).

SUGGESTION. Prove the sum equal to an altitude of the triangle.

34. In any triangle the sum of two sides is greater than twice the median on the third side.

E

F

35. ABCD is a square, and EFGH a rectangle. Does it follow that AAEHA FCG and ▲ EBFA HDG? Prove.

G

B

36. If a median of a triangle is equal to half the base, the vertex angle is a right angle.

37. State and prove the converse of the theorem in Ex. 36.

CHAPTER II.

STRAIGHT LINES AND CIRCLES.

180. A circle (§ 12) divides the plane into two parts such that any point which does not lie on the circle lies within it or outside it.

181. A line-segment joining any two points on a circle is called a chord. A chord which passes through the center is a diameter.

182. If a chord is extended in one or both directions, it cuts the circle and is called a secant.

Outside Point

183. A tangent is a straight line which touches a circle in one point but does not cut it. An indefinite straight line through a point outside a circle is a secant, a tangent, or does not meet the circle.

184. The portion of a circle included between any two of its points is called an arc (§ 12). An arc AB is denoted by the symbol AB.

Tangent

Secant

A circle is divided into two arcs by any two of its points. If these arcs are equal, B each is a semicircle. Otherwise one is called the major arc and the other the minor arc.

arc.

Major Arc

Unless otherwise indicated AB means the minor
In case of ambiguity a third letter may be used, as arc AmB.

An arc is said to be subtended by the chord which joins its end-points. Evidently every chord of a circle subtends two arcs. Unless otherwise indicated the arc subtended by a chord means the minor arc. 185. An angle formed by two radii is called a central angle. An angle formed by two chords drawn from the same point on the circle is called an inscribed angle.

If the sides of an angle meet a circle the arc or arcs which lie within the angle are called intercepted arcs.

If the vertex of the angle is within or on the circle there is only one intercepted arc; if it is outside the circle there are two intercepted arcs, as AB and CD in the figure.

186. If a circle is partly inside and partly outside another circle, then they cut each other.

If two circles meet in one and only one point, they are said to be tangent.

Arcs of two circles are tangent to each other if the complete circles of which they form a part are

tangent to each other.

187. Two circles which can be made to coincide are said to be equal.

The word congruent is unnecessary here, since all circles are similar.

1. Does the word circle as used in this book (§ 12) mean a curved line or the part of the plane inclosed by that line?

2. In how many points can a straight line cut a circle?

3. In how many points can two circles cut each other?

PRELIMINARY THEOREMS ON THE CIRCLE.

189. Radii or diameters of the same circle or of equal circles are equal.

190. If the radii or diameters of two circles are equal, the circles are equal.

191. A diameter of a circle is double the radius.

192. A point lies within, outside, or on a circle, according as its distance from the center is less than, greater than, or equal to the radius.

193. If an unlimited straight line contains a point within a circle, then it cuts the circle in two points.

194. If two circles intersect once, they intersect again. See figure, § 186.

195. If a straight line is tangent to each of two circles at the same point, then the circles do not intersect, but are tangent to each other at this point. See § 186.

196. If two arcs of the same circle or equal circles can be so placed that their end-points coincide and also their centers, then the arcs coincide throughout or else form a complete circle.

197. If in two circles an arc of one can be made to coincide with an arc of the other, the circles are equal. 198. A circle is conveniently referred to by indicating its center and radius.

Thus, OA means the circle whose center is O and radius OA. When no ambiguity arises, the letter at the center alone may be used to denote the circle. Thus, OC means the circle whose center is C.