23. To describe a rectangle, whose length and breadth shall be equal to two given lines, A B and C.

At the point B, in the given line A B, erect the perpendicular B D, and make it equal to C: from the point D A, with the radii A B and C, describe two arcs cutting each other in E, then join E A and E D, and A B, D E, will be the rectangle required.

E

Fig. 23.

24. Upon a given line to describe a rectangle that shall be equivalent to a given rectangle.

Let A D be the line, and A B, F C, the given rectangle. Find a fourth proportional to the three lines, AD, A B, A C, and let

Fig. 24.

F

A

Ax be that fourth proportional: a rectangle constructed with the lines A D, and Ax will be equivalent to the rectangle ABF C.

25. The two diagonals of a parallelogram bisect each other.

In the parallelogram A B C D, the diagonals A C, B D, bisect each other in the point O, because A B, B D, meet the parallel right lines A D, B C; the angles AL

Fig. 25.

OA D, O D A, are respectively equal to O C B, O B C, and A D being equal to B C, the triangle O A D, O C B, have the sides O A, O D, respectively equal to O C, O B, and therefore A C, B D, are bisected at the point O.

26. The square described on the hypotenuse of a right-angled triangle is equivalent to the sum of the squares described on the other two sides. (Fig. B)

Fig. A. Let the triangle A B C be right-angled at A. Having described squares on the three, sides, let fall from A, on the hypotenuse, the perpendicular A D, which produce to E; and draw the diagonals A F, CH. The angle A B F is made up of the angle A B C, together with the right angle C B F: the angle C B H is made up of the same angle A B C, together with

the right angle A B H; hence the angle A B F is equal to HB C. But we have A B equal to B H, being sides of the same square B F, equal to B C, for the same reason; therefore the triangles A B F, H B C, have two sides, and the included angle in each equal; consequently they are themselves equal. The triangle A B F is half of the rectangle B E, because they have the same base B F, and the same altitude B D. (Cor. 1.) The triangle H B C is in like manner half of the square A H, for the angles BA C, B A L, being both right angles, A C and A L form one and the same straight line parallel to H B; consequently the triangle H B C, and the square A H, which have

the common base B H, have also the common altitude A B, hence the triangle is half of the square. The triangle A B F has been proved equal to the triangle H B C, hence the rectangle BDEF, which is double of the triangle A B F, must be equivalent to the square A H, which is double of the triangle H B C. In the same manner it may be proved that the rectangle C D E G is equivalent to the square A F. But the two rectangles BDE F, C DE G, taken together, make up the square B C G F, therefore the square B C G F, described on the hypotenuse, is equivalent to the sum of the squares A BHL, ACG K, described in the other two sides; in other words, (Fig. B) the square A contains just as many square feet or yards as are contained in the other two squares B and C.

SECTION 4.- -THE CIRCLE AND MEASUREMENT OF ANGLES. 27. The circumference of a circle is a curved line, all the points of which are equally distant from a point within, called

Fig. 27.
H

F

G

the centre. Every straight line, CA, C E, C D, drawn from the centre to the circumference is called a radius, or semi-diameter; every line which, like A B, passes through the centre, and is terminated on both sides by the circumference, is called a diameter; it thus follows, that all the radii are equal, and all the diameters are equal also, and each double of the radius. A The arc is a part of the circumference FH G.

The chord of an arc is the straight line F G.

E

D.

B

The segment is the surface or portion of a circle included between an arc and its chord.

28. An inscribed triangle is one which, like BA C, has its three angular points in the circumference.

29. An inscribed angle is one which has its A vertex in the circumference, and is formed by two chords BA C. In general, an inscribed figure is one of which all the angles have their vertices in the circumference.

Fig. 29.

30. A secant is a line which meets the circumference in two points, and lies partly within and partly without the circle. A Bis

a secant.

C

m

D

A tangent is a line which has but one line. in common with the circumference. C D is a tangent. The point m, where the tangent c touches the circumference, is called the point of contact. Two circumferences touching each other have one point in common.

31. To divide a given circle into any proposed number of parts that shall be equal to each other, both in area and perimeter.

Anal Fig. 31.

32. To divide a given circle into any number of equal parts by means of concentric circles.ade i o

circle; also divide O HA
into as many equal
parts as there may be
required; then draw
perpendiculars from
the points of division
to the semicircle on
OH; then with the
centre H on the radii
Hb, H c, &c., de-

scribe circles, and the ab

2

33. Every diameter divides the circle and its diameter into two

equal parts.

Let A E B F be a circle, and A B a diameter: now if the figure A E B be applied to A F B, their common base A B retaining its position, the curve line A E B must fall exactly on the curve line A F B, A otherwise there would, in the one or the other, be points unequally distant from the centre, which is contrary to the definition of a circle.

Fig. 33.

F

C

34. If the distance between the centre of two circles is equal to the sum of their radii, the two circles will touch each other externally.

centres must be less than the sum of their radii.

35. In the same circle, or in equal circles, equal angles having their vertices at the centre intersect equal arcs on the circumference; and conversely, if the arcs intercepted are equal, the angles contained by the radii will also be equal.

Let C and C be the vertices of equal angles, and the angles A C B equal DCE; since the angles AC B, D C E, are equal, they may be placed upon each other; and since their sides are equal, the point A will evidently fall on D, and the point B on E;

but in that case, the arc A B must also fall on the arc D E; for if the arcs did not exactly coincide, there would, in the one or the other, be points unequally distant from the centre; which is impossible; hence the arc A B is equal to D E.