which he is standing, and ✪ the altitude of the sun, the height of the pyramid will be

tan √(10a2 – 4ab + 2b2).

23. A, B, C are three points in a horizontal plane forming an isosceles triangle right-angled at C, and AB= c. At A, B, C the angles of elevation of an object P are a, a, ẞ respectively. Find h and show that ẞ cannot be less than the angle whose

tangent is

24.

tan a

√2

Three mountain peaks A, B, C appear to the observer to be in a straight line, when he stands at each of two places P and Q in the same horizontal line. The angle subtended by AB and BC at each place is a; and the angles AQP, CPQ are and y.

Prove that the heights of the mountains are as

cot 2a + cot: tan a (cot a + cot ¥) (cot a + cot 4) : cot 2a + cot &; and that if QB cut AC in D,

AC=CDx sin 2a (cot a + cot ¥).

25. A man standing on a plane observes a row of equal and equidistant pillars, the 10th and 17th of which subtend the same angle as they would if they were in the position of the first and were respectively and of the height. Show that, neglecting the height of the eye, the line of the pillars is inclined to the line drawn to the first at an angle whose secant is nearly 2·6.

CHAPTER IX.

THE GEOMETRY OF THE TRIANGLE.

182. In this chapter will be given the most fundamental propositions on the circles and centres of a triangle.

For the sake of avoiding repetition, the following mode of lettering will be consistently adopted. (Explanation of the terms will be given in the course of the chapter.)

The triangle considered ABC. The middle points of the sides D, E, F. The feet of the perpendiculars L, M, N. The points of contact with the inscribed circle X, Y, Z. The centre of the inscribed circle, I. The centre of the circumscribed circle, S. The centre of gravity, G. The orthocentre, O. The centre of the cosine circle, K. The centre of the nine-points circle, T. The centre of the Lemoine and Brocard circles, V. The middle points of AO, BO, CO; P, Q, R. The centres of the escribed circles I, 12, 13. Their points of contact with the sides X1, Y1, Z1, &c.

Antiparallels.

183. DEF. Two lines are said to be antiparallel with respect to any angle, when the inclination of one to one of the lines containing the angle is equal and opposite to the inclination of the other to the other of the lines containing the angle.

Thus, in the figure, BC, B'C' are antiparallel with respect to the angle A:-the angles ABC, AB'C' being equal and opposite* to one another.

It follows that the angles C'BC, C'B'C are together equal to two right angles; and, therefore, the quadrilateral BCB'C' is inscribable in a circle. Hence antiparallels might be thus defined :—

Two opposite sides of a quadrilateral inscribable in a circle are said to be antiparallel with respect to the angle contained by the other two opposite sides.

The following statements are obvious :—

Lines which are, with respect to any angle, antiparallel to the same line are parallel to one another.

Through any point, one and only one line can be drawn which is, with respect to any angle, antiparallel to another line. These statements are sufficient to indicate the analogy between antiparallels and parallels.

* That is, BC has revolved from BA in the opposite direction to that in which B'C' has revolved from B'A.

184.

Another mode of defining antiparallels is the following:

Two lines are said to be antiparallel with respect to two other lines when the bisectors of the angles between the first two are in the same directions as the bisectors of the angles between the second two.

Thus, in the figure, the lines BC, B'C' are antiparallel with respect to the lines BC', B'C; for the bisector of the obtuse angle between BC and B'C' is in the same direction as the bisector of the acute angle between BC' and B'C': and the bisector of the acute angle between BC and B'C' is in the same direction as the bisector of the obtuse angle between BC' and B'C.

This will become clear if parallels to the four lines BC, B'C', BC', B'C be drawn through any point 0.

It is clear that the relations between the pair BC, B'C' and the pair BC', B'C are reciprocal.

It should be observed that, if the angles ABC, ACB are equal, the antiparallels to BC with regard to A become parallels to BC.

185. The following proposition is important:

A line, drawn from the vertex of an angle, cuts any two parallel intercepts of the angle in the same ratio.

This follows at once from Euc. VI. 4.

In particular:-The line, drawn from the vertex of a triangle to bisect the base, bisects all the parallels to the base.

When an antiparallel to a side of a triangle is spoken of, the antiparallel must be understood to be drawn with respect to the angle opposite that side.

Thus, in considering the triangle ABC of the figure of Art. 183, B'C' is called an antiparallel to the side BC' (without specifying the angle A).

Since the antiparallels to a side are parallel to one another, the line from the opposite angle bisecting one of them will bisect all.

Centres of Similitude.

186. DEF. The point which divides the distance between the centres of two circles internally (or externally) in the ratio of the radii of the circles is called the internal (or external) centre of similitude of the two circles.

PROP. The line joining the extremities of any two parallel radii of two circles cuts the line of their centres in one or other of the centres of similitude.

For, let 0, be the centres of two circles, and OR, or two parallel radii. Let Rr cut Oo in S. Then the triangles ROS, roS are equiangular;

:. OS: OSOR : or,

:. S is a centre of similitude.

If Rr is a common tangent of the two circles, OR and or are both at right angles to Rr; and are, therefore, parallel: hence, a common tangent to two circles cuts the line of their centres in one or other of the centres of similitude.

COR. If the circles do not cut, there is a pair of common tangents through each centre of similitude, the angle between which is bisected by the line of centres.

Hence the two common tangents through one centre of similitude are antiparallel with respect to the angle formed by the two common tangents through the other centre of similitude.

Pole and Polar.

187. DEF. The point of intersection of the tangents to a circle drawn from the extremities of any chord is called the pole of the chord and the chord is called the polar of the point of intersection.