112. Two triangles are similar if two angles of the one are equal respectively to two angles of the other.

113. The lines bisecting the angles of a parallelogram form a rectangle whose diagonals are parallel respectively to the sides of the parallelogram.

114. If two triangles have one angle equal, and a second angle supplementary, the sides including their third angles have the same ratio. (Place the equal angles on each other.)

115. If two triangles have one angle equal and the sides about a second angle have the same ratio, their remaining angles are either equal or supplementary.

If in each triangle the side opposite the given equal angle is greater than the side adjacent, or if the given angle is not acute, or if their third angles are both acute, or both obtuse, the triangles are similar. Compare I. 96-100.

116. Two triangles having an angle of the one equal to an angle in the other are to each other as the rectangles of the sides containing the equal angles; or (Fig. Art. 50)

first producing FA to GH, and producing GH, KI, and LB till

they meet.

120. Prove Theorem XXVII., first constructing the squares on opposite sides of A B and B C from that on which they are drawn in the figure in Art. 117; moving the square A G H B on A B, a distance equal to BC in the direction BA; then proving that these squares are divided into parts that can be made to coincide with the parts of the square on A C

121. In the figure in Art. 117 draw HI, KE, F G. The triangle HIB is equal, and the triangles CKE, GAF are equivalent to ABC.

122. If G F and K E are drawn, G F2 + K E2 = 5 A C2.

From G and K draw perpendiculars to FA and E C respectively. (70.)

123. If from any point P straight lines are drawn to the vertices of a rectangle A B C D, P A2 + P C2 = P B2 + P D2.

124. Prove Theorem XXVIII. by means of (90).

125. Prove Theorem XXIX. by means of (90).

126. In the Fig. in Art. 86, if A D, BE, CF bisect respectively BC, CA, A B,

1st. If A DCF, A B C is isosceles.

2d. Any line G H drawn from A B to A C parallel to BC is
bisected by A D.

3d. If G C and H B are drawn they will intersect in A D.
4th. The triangle constructed with the sides AD, BE, CF,
is to A B C as 3: 4.

5th. 4 (A D2+BE2 + C F2) = 3 (A B2 + B C2 + C A2).

127. The squares of the sides of a right triangle are as the segments of the hypothenuse made by a perpendicular from the vertex of the right angle to the hypothenuse.

128. The square of the hypothenuse is to the square of either side as the hypothenuse is to the segment adjacent to this side made by a perpendicular from the vertex of the right angle.

129. The side of a square is to its diagonal as 1:2; or the square described on the diagonal of a square is double the square itself.

BOOK III.

THE CIRCLE.

DEFINITIONS.

1. A Circle is a plane figure bounded by a curved line called the circumference, every point of which is equally distant from a point within called the centre; as A B D E.

2. An Arc is any part of the circumference; as AF B.

3. A Chord is the straight line joining the ends of an arc; as A B.

4. The Diameter of a circle is a chord passing through the centre; as A D.

B

F

E

G

5. The Radius of a circle is a line drawn from the centre to the circumference; as CD.

6. Corollary. The radii of a circle, or of equal circles, are equal; also the diameters are equal, and each is equal to double the radius.

7. A Segment of a circle is the part of the circle cut off by a chord; as the space included by the arc A FB and the chord А В.

8. A Sector is the part of a circle included by two radii and the intercepted arc; as the space B C D.

9. A Tangent (in geometry) is a line which touches, but does not, though produced, cut the circumference; as G D.

A tangent is often considered as terminating at one end at the point of contact, at the other where it meets another tangent or a secant.

10. A Secant (in geometry) is a line lying partly within and partly without a circle; as G E.

A secant is generally considered as terminating at one end where it meets the concave circumference, and at the other where it meets another secant or a tangent.

THEOREM I.

11. In the same circle, or equal circles, equal angles at the centre are subtended by equal arcs; and, conversely, equal arcs subtend equal angles at the centre.

G

H

Place the angle B on the angle E; as they are equal they will coincide; and as BA and BC are equal to ED and E F, the point A will coincide with D, and the point C with F; and the arc AC will coincide with DF, otherwise there would be points in the one or the other arc unequally distant from the centre.

Conversely. If the arcs AC and D F are equal, the angles B and E are equal.

For, if the radius AB is placed on the radius DE with the point B on E, the point A will fall on D, as AB= DE; and the arc AC will coincide with D F, otherwise there would be points in the one or the other arc unequally distant from the centre; and as the arc A C D F, the point C will fall on F; therefore BC will coincide with EF, and the angle B be equal to E.

THEOREM II.

12. In the same circle, or equal circles, equal arcs are subtended by equal chords; and conversely, if the chords are equal, the arcs are equal.

placed on the centre of D E F with the point A of the circumference on the point D, as the arcs are equal, B will fall on E, and the chord AB will coincide with DE; therefore AB=DE. Conversely. If the chords A B and D E are equal, the arcs A B and D E are equal.

Draw the radii G A, G B, HD, HE. The triangles A B G, DEH, being mutually equilateral, are equal (I. 88), and the angles at G and H are equal; hence (11) the arcs A B and DE are equal.

THEOREM III.

13. The radius perpendicular to a chord bisects the chord and the arc subtended by the chord.

Let CE be a radius perpendicular to the chord A B; it bisects the chord A B, and also the arc A E B.

Draw the radii CA and C B and the chords A E and E B. As equal oblique

A

B

lines are equally distant from the perpendicular, A D = DB (I. 92); and as

E

E is a point in the perpendicular to the middle of A B, it is equally distant from A and B (I. 94); therefore the chords and hence (12) the arcs A E, E B are equal.