9. Parallelograms of equal bases are as their heights.

10. Parallelograms are to one another, as their bases and heights.

11. In any parallelogram the sum of the squares of the diagonals is equal to the sum of the squares of all the four

sides.

12. The sum of the four internal angles of any quadrilateral figure, is equal to four right angles.

13. If two angles of a quadrangle be right angles, the sum of the other two amounts to two right angles.

14. The sum of all t. e internal angles of a polygon is equal to twice as many right angles, abating four, as the polygon has sides.

15. Hence all right-lined figures of the same number of sides, have the sum of all the internal angles equal.

16. The sum of the external angles of any polygon is equal to four right angles.

17. All right-lined figures have the sum of their external angles equal.

D
E

P

AØ

T

R

18. In two similar figures a C, P R; if two lines B E, Q T, be drawn after a like manner, as suppose, to make the angle C BER QT; then these lines have the same proportion, as any two homologous sides of the figure; B VİZ. BEQT :: B C : Q R :: A B : P Q :: A D¦ P s.

19. All similar figures are to one another as the squares of their homologous sides.

21. Any regular figure A B C D E, is equal to a triangle whose base is the perimeter ABC DE A; and height, the perpendicular o P, drawn from the centre, perpendicular to one side.

22. Only three sorts of regular figures can fill up a plane surface, that is, the whole space round an assumed point; and these are six triangles, four squares, and three hexagons.

SECTION IV. Of the Circle, and Inscribed and Circum scribed Figures.

Definitions.

1. A circle is a plane figure described by a right line moving about a fixed point, as A c about c or it is a figure bounded by one line equidistant from a fixed point.

2. The centre of a circle is the fixed point about which the line moves, C.

3. The radius is the line that describes the circle, c a. Cor. All the radii of a circle are equal.

4. The circumference is the line described by the extreme end of the moving line, A B D A.

B

G

5. The diameter is a line drawn through the centre, from one side to the other, A

D

A D.

E

6. A semicircle is half the circle, cut off by the diameter, as

A B D.

7. A quadrant, or quarter of a circle, is the part between two radii perpendicular to one another, as C D E.

8. An arch is any part of the circumference A B.

9. A sector is a part bounded by two radii, and the arch between them, A C B.

10. A segment is a part cut off by a right line, or cord, DE F, or D A B F.

11. A cord, a right line drawn through the circle, as D F.

12. Angle at the centre is that whose angular point is at the centre A c B. (See the last figure.)

C

13. Angle at the circumference is when the angular point is in the circumference,

as BA D.

B

C

14. Angle in a segment, is the angle made by two lines drawn from some point of the arch of that segment to the ends of the base; as B C D is an angle in the segment B C D.

15. Angle upon a segment, is the angle made in the opposite segment, whose sides stand upon the base of the first; as BA D, which stands upon the segment B C D.

16. A tangent is a line touching a circle, which produced, does not cut it, as G A F. (Fig. to def. 5.)

17. Circles are said to touch one another, which meet, but do not cut one another.

18. Similar arches, or similar sectors, are those bounded by radii that make the same angle.

19. Similar segments are those which contain similar triangles, alike placed.

20. A figure is said to be inscribed in a circle, or a circle circumscribed about a figure, when all the angular points of the figure are in the circumference of the circle.

21. A circle is said to be inscribed in a figure, or a figure circumscribed about a circle, when the circle touches all the sides of the figure.

22. One figure is inscribed in another, when all the angles of the inscribed figure are in the sides of the other.

Prop. 1. The radius c R, bisects any cord at right angles, which passes not through the centre, as A B.

R

Cor. 1. If a line bisects a cord at right angles, it passes through the centre of the circle.

Cor. 2.

arch.

The radius that bisects the cord also bisects the

2. In a circle equal cords are equally distant from the

centre.

3. If several lines be drawn through a circle, the greatest is the diameter, and those that are nearer the centre are greater than those that are farther off.

4. If from any point three equal right lines can be drawn to the circumference; that point is the centre.

5. No circle can cut another in more than two points

6. There can only two equal lines be drawn from any exterior point P, to the circumference of a circle.

7. In any circle, if several radii be drawn making equal angles, the arches and sectors comprehended thereby will be equal, if ACB=B CD: then, arch A B=arch BD; and sector A CB=BC D.

D

8. In the same or equal circles, the arches, and also the sectors, are proportional to the angles intercepted by the radii. 9. The circumferences of circles are to one another as their diameters.

10. A right line, perpendicular to the diameter of a circle, at the extreme point, touches the circle in that point, and lies wholly without the circle.

11. If two circles touch one another either inwardly or outwardly, the line passing through their centres shall also pass through the point of contact.

12. In a circle the angle at the centre is double the angle at the circumference, standing upon the same arch; BD C 2 BAC.

B

13. All angles in the same segment of a circle are equal, D A C=D B C, and D G C=D H C.

G

14. If the extremities of two equal arches D A, B c, be joined

by right lines, DC, A B; they will be parallel.

B

F

15. The angle A B C in a semicircle is a

right angle.

D

16. The angle AB G, in a greater segment A B F G, is less than a right angle; and the angle A B F, in a less segment A B F, is greater than a right angle.

Then arch B D + GH = 2 E F,.if A is within the circle; or arch B D - GH = 2 E F, if A is without.

18. If from a point without, two lines touch a circle: the angle made by them is equal to the angle at the centre, standing on half the difference, of these two parts of the circumference.

19. The angle A = BHD. G

+ HDG, when A is within; or
A = B HD — H D G, when a is
without the circle.

H

20. In a circle, the angle made at the point of contact between the tangent and any chord, is equal to the angle in the alternate segment; ECF E B c, and E CA

= EG C.

B

H

D

A

G

B

21. A tangent to the middle point of an arch, is parallel to the chord of it.

22. If from any point в in a semicircle, a perpendicular BD be let fall upon the diameter, it will be a mean proportional between the segments of the diameter;

ADD BD BD C.

A

B

D

B

23. The chord is a mean proportional between the adjoining segment and the diameter, from the similarity of the triangles: that is, A D A B ::AB: AC; and CD: CB CB: CA. 24. In a circle if the diameter A D be drawn, and from the ends of the cords A B, A C, perpendiculars be drawn upon the diameter; the squares of the chords will be as the segments of the diameter; AE: AFA B : A c2.

25. If two circles touch one another in P, and the line PD E be drawn through their centres; and any line P A B is drawn through that point to cut the circles, that line will P be divided in proportion to the diameters;

PA: PB: PD: PE.

A

E F

A

D

D