26. If from two fixed points in the circumference of a circle, straight lines be drawn intercepting a given arc and meeting without the circle, the locus of their intersections is a circle.

27. Show that, if two circles cut each other, and from any point in the straight line produced which joins, their intersections two tangents be drawn, one to each circle, they shall be equal to one another.

28. If three equal circles have a common point of intersection, prove that a straight line joining any two of the points of intersection will be perpendicular to the straight line joining the other two points of intersection.

29. Two equal circles are drawn intersecting in the points A and B, a third circle is drawn with centre A and any radius not greater than AB intersecting the former circles in D and C. Show that the three points B, C, D, lie in one and the same straight line.

30. Through two given points to describe a circle bisecting the circumference of a given circle.

31. If two circles touch each other internally, prove that the straight lines which join the extremities (on the same side of the common diameter) of any two parallel diameters, pass through the point of contact.

32. A common tangent is drawn to two circles which touch each other externally; if a circle be described on that part of it which lies between the points of contact, as diameter, this circle will pass through the point of contact of the two circles, and will touch the line which joins their centres.

33. If two circles touch each other externally, and parallel diameters be drawn, the straight line joining the extremities of these diameters will pass through the point of contact.

34. If two circles touch each other, any straight line passing through the point of contact cuts off segments which contain equal angles.

35. If a circle roll within another of twice its size, any point in its circumference will trace out a diameter of the first.

36. Given two circles: it is required to find a point from which tangents may be drawn to each, equal to two given straight lines.

37. Shew that all equal straight lines in a circle will be touched by another circle.

38. Two circles are described about the same centre: draw a chord to the outer, which shall be divided into three equal parts by the inner one. How is the possibility of the problem limited? 39. The circles described on the sides of any triangle as diameters will intersect in the sides, or sides produced, of the triangle.

40. The circles which are described upon the sides of a rightangled triangle as diameters, meet the hypothenuse in the same point; and the line drawn from the point of intersection to the centre of either of the circles will be a tangent to the other circle.

Arithmetical Exercises.

1. Two circles intersect, and lines are drawn cutting them, one of which passes through the centres, and cuts the other in the middle point O of the chord of contact. The parts of the line not passing through the centres are E F=3} ft., FO=4}, OG =3, GH=3; and of the other, A B not measured, BO=3, OC=2, CD not measured. Find the lengths of the radii of the circles, and the chord of contact. Ans. 6 ft., 5% ft., 10 ft.

2. In a circle of radius 10 inches, a square is inscribed, and in the square a circle. Find the length of the side of the square and the radius of the inner circle.

Ans. 14'142136 in. and 7.071068 in. 3. A flag-staff, 45 ft. high, is erect on a tower 90 ft. high; at what point on the horizon will the flag-staff appear under the greatest angle? Ans. 110 227 ft.

4. A person observes the elevation of a tower to be 60°, and in receding from it 100 yards farther he finds the elevation to be 30°. Required the height of the tower. Ans. 86.754 ft.

5. An equilateral triangle is described having its angular points in the sides of a right-angled isosceles triangle, and one side parallel to the hypothenuse. The length of a side of the rightangle is 10 inches; find the the area of the equilateral triangle. Ans. 44 8038 sqr. in.

6. Two objects, A and B, were observed from a ship to be at the same instant in a line inclined at an angle of 15° to the east of its course, which was at the time due north. The ship's course was then altered, and after sailing five miles in a N.W. direction, the same objects were observed to bear E. and N.E. respectively. Required the distance of A from B.

Ans. 6'33974 miles.

CHAPTER XI.

SIMILAR FIGURES.

265. When a design is copied on a reduced scale by the methods described in § 185-9 the drawing produced is said to be similar to the

original. The conditions that the one shall be a true plan of the other are, 1st, that the lines of the copy shall have to one another the same ratio as the corresponding lines of the original; and 2nd, that the angles of the one shall be equal to those of the other. The corresponding lines of two figures are said to be homologous.

For the sake of simplicity, let us consider only figures composed of straight lines, as ABCD, A'B'C'D' (fig. 265). The resemblance or similitude of these figures may be defined as follows.

1wo figures are similar when the angles of the one are equal to those of the other, each to each, and their homologous sides are proportional.

B'

B

D'

C'

Fig. 265.

The constant ratio of the homologous lines is termed the ratio of similitude of the figures.

Of geometrical figures formed by straight lines the triangle is the simplest, and every other rectilinear figure can be divided into triangles, so that it will be convenient to consider first the conditions of similarity of triangles. In similar triangles, the homologous sides are those opposite to the equal angles.

Two triangles are similar when the angles of one are equal to those of the other, each to each. 266. Let A B C, A'B'C' be the triangles (fig. 266). And let the angle

cides with its equal A, then the side B'C' will have a direction D E parallel to B C, since the corresponding angles AD E, A B C are equal. ThereforeAD A E DE

Therefore two triangles which have their angles equal have their homologous sides proportional, and are consequently similar.

Two triangles which have their sides parallel or perpendicular, each to each, are similar.

267. It is sufficient to remark, that the angles of the one are respectively equal to those of the other. Thus, in the triangles A B C, A'B'C' (fig. 267, the angle A A', since their sides are parallel and are drawn in the same direction, and for similar reasons the angle B = B', therefore the triangles are similar.

In the triangles A B C, A'B'C' (fig. 268), which have their sides perpendicular, the angle A=A', B=B', therefore the two triangles are similar.

Two triangles which have their sides proportional are similar.

268. Let ABC, A'B'C' be the triangles. On AB take AD equal to A'B', and on AC take AE equal to

A'C'; the triangle ADE will be similar to ABC.

But since by hypothesis, the triangles A B C, A'B' C′ have their sides proportional, therefore

Comparing these proportions, since ADA'B', we see that A E must be equal to A'C', and that DE must be equal to B'C'. Hence the triangle A'B'C' is similar to A D E, and therefore to A B C.

Two triangles which have two sides of the one proportional to two sides of the other, and the included angles equal, are similar.

269. Let the angle A=A', and let

AB

AC

A'B'

A'C'

The

A C

AE

Take A D equal to A'B', and A E equal to A'C'.

triangle A D E equals A'B'C'. But since

therefore D E is parallel to B C. Hence the triangle ADE, or its equal A'B'C', is similar to AB C.

On a given straight line to construct a triangle similar to a given triangle.

270. Let ABC (fig. 270) be the given triangle ; make