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I 24. Find the angles subtended at the centre of a circle by the three segments made by the sides of the circumscribed square on any tangent to the circle.

125. With three given points as centres to describe three circles touching each other. Shew that there are, in general, four solutious.

126. If the straight lines which bisect the angles of a rectilineal figure all pass through one point, a circle may be inscribed in the figure.

127. Shew that a regular polygon has, or has not, a centre of symmetry, according as the number of its sides is even

or odd.

128. A regular polygon of n sides has n or an axes of symmetry, according as n is odd or even.

129. Describe a circle to touch each of two given lines, and having its centre at a given distance from a third given line. How many

such circles can be described ? 130. Find a point O within a given triangle such that, if AO, BO, CO be joined, the angles OBC, OCA, OAB shall all be equal to one another.

131. If the centres of the inscribed and circumscribed circles of a triangle coincide, the triangle must be equilateral.

132. A circle is inscribed in an isosceles triangle. Shew that the triangle formed by joining the points of contact is also isosceles. 133

The locus of the centre of the inscribed circle of a triangle, whose base and vertical angle are given, consists of two circular arcs.

134. If DA be one side of a regular hexagon in a circle, AB a tangent equal to it and making an obtuse angle with it, C the centre of the circle, and if BD meet the circle in E and BC meet the nearer part of the circumference in F, prove that AE and EF are equal to sides of regular figures of twelve and twenty-four sides respectively inscribed in the same circle.

135: Construct an isosceles triangle having given its base equal to the greater, and the diameter of its inscribed circle equal to the less, of two given straight lines.

136. In a given right-angled triangle the sides containing the right angle are 6 and 8 respectively. Find the lengths of the segments into which the hypotenuse is divided by the inscribed circle.

13.7 Construct a triangle having given the vertical angle and the segments of the base made by the inscribed circle.

138. A square and an equilateral triangle are inscribed in the same circle. Prove that the area of the square is twothirds of the square on the side of the triangle.

SECTION VIII.

THE CIRCLE IN CONNECTION WITH AREAS.

THEOR. 28. If a chord of a circle is divided into two segments by a point in the chord or in the chord produced, the rectangle contained by these segments is equal to the difference of the squares on the radius and on the line joining the given point with the centre of the circle.

Let BC a chord of a circle whose centre is O be, divided into two segments at the point A:

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then shall the rectangle contained by AB and AC be equal to the difference of the squares on OB and OA.

III. 9.

Draw OD perpendicular to BC; then BD is equal to DC. Hence AC is equal to the sum, and AB to the difference, of BD and AD; therefore the rectangle contained by AB and AC is equal to the difference of the squares on BD and AD.

II. 8.

ll. 9.

II. 9.

But the square on OB is equal to the sum of the squares on BD and OD, and the square on OA is equal to the sum of the squares on AD and OD; therefore the difference of the squares on OB and OA is equal to

Ax. e. the difference of the squares on BD and AD. Hence the rectangle contained by AB and AC is equal to the

Ax. C. difference of the squares on OB and OA.

Q.E.D.

COR. 1. The rectangle contained by the segments of any chord passing through a given point is the same whatever be the direction of the chord.

COR. 2. If the point is within the circle, the rectangle contained by the segments of any chord passing through it is equal to the square on half that chord which is bisected by the given point. [Particular case of Cor. 1.)

COR. 3. If the point is without the circle, the rectangle contained by the segments of any chord passing through it is equal to the square on the tangent to the circle drawn from that point. [Particular case of Cor. 1.]

COR. 4.

Conversely, if the rectangle contained by the segments of a chord passing through an external point is equal to the square on the line joining that point to a point in the circumference of the circle, this line touches the circle.

For, if the rectangle contained by AB, AC is equal to the square on AP, and AP does not touch the circle, AP will meet the circle again in Q; therefore, by Cor. 1, the rectangle contained by AP, AQ is equal to the rectangle contained by AB, AC, which by hypothesis is equal to the square on AP; but this is impossible ; therefore AP touches the circle at P.

Ex. 139. Conversely to Cor. 2, shew that if the rectangle contained by the segments of a chord passing through a point within the circle is equal to the square on a line joining that point to a point in the circumference of the circle, this line produced through the point to meet the circumference will be bisected at the point.

Ex. 140. A circular arch of 60 feet span is 18 feet high : find the radius of curvature of the arch.

Ex. 141.

If the water flowing through this arch were to rise 14 feet, by how much would the span be diminished ?

Ex. 142. An arch is to be built so as to give a roadway 60 feet wide and a pathway on each side 10 feet wide with a height at the curb of the path of 1o feet; find the radius of curvature and height of the arch.

Ex. 143. Assuming that an eye six feet above the level of a still lake can just see a light also six feet above the level of the lake at a distance of six miles : find the diameter of the earth in miles.

Ex. 144.

From the data of the last question, shew that if h be the height of the eye above the sea in feet, and d the distance of the offing in miles, d=vh.

Ex. 145. The lantern of a lighthouse is 96 feet above the sea level, to what distance does it illuminate the sea ? and how far would it be visible to an eye 24 feet above the surface ?

Ex. 146. What is the extreme distance at which two mountain summits, each 1,350 feet high, would be visible to one another ?

Ex. 147. Shew why a ship sailing away from the land disappears more and more rapidly from a spectator on the land, as she gets farther away.

Ex. 148. Compare the diameters of two globes, on one of which objects of a given height are just visible to one another at double the extreme distance at which they are so on the other.

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