EXERCISES ON BOOK III.

*165. If the perpendicular AD on the side BC of the triangle ABC meets the circumscribed circle of the triangle at P, and if either of the perpendiculars from B or C on the opposite side of the triangle meet AD at O; shew that OD is equal to DP.

*166. The perpendiculars from the vertices of a triangle on the opposite sides meet in a point (called the orthocentre of the triangle).

*167. If O is the orthocentre of the triangle ABC, then any one of the four points O, A, B, C is the orthocentre of the triangle whose vertices are the other three.

*168. The rectangles contained by the segments into which the orthocentre divides each of the perpendiculars from the vertices of a triangle on the opposite sides are equal to one another.

*169. If three straight lines drawn from the vertices of a triangle to the opposite sides meet in a point and are divided at that point into segments containing equal rectangles, shew that the point must be the orthocentre of the triangle.

*170. If O is the orthocentre of the triangle ABC, the angles BOC, COA, AOB are either supplementary or equal to the angles A, B, C respectively.

*171. If any one of four points be the orthocentre of the triangle formed by the other three, shew that the circles through any three of the points are equal to one another.

*172. The distances of the orthocentre of the triangle

ABC from A, B, C are respectively double of the distances of the centre of the circumscribed circle from BC, CA, AB.

*173. AD, BE, CF are drawn perpendicular to the sides BC, CA, AB of a triangle ABC. If the triangle is acuteangled shew that these perpendiculars bisect the angles of the triangle DEF. What is the corresponding theorem when the triangle is obtuse-angled?

*174. If O is the orthocentre of an acute-angled triangle ABC, D, E, F the feet of the perpendiculars from the vertices on the opposite sides, shew that O is the centre of the inscribed. and A, B, C the centres of the escribed circles of the triangle DEF

*175. The circle through the middle points of the side: of a triangle ABC also passes through the feet of the perpendicu lars from the vertices on the opposite sides. Hence if O be the orthocentre the same circle also passes through the middle poin's of OA, OB, OC. (The circle is called the nine-points circle of the triangle.)

*176. The centre of the nine-points circle of a triangle bisects the straight line joining the orthocentre and the centre of the circumscribed circle.

*177. The diameter of the nine-points circle of a triangle is half that of the circumscribed circle.

178. Find the locus of a point from which equal tangents can be drawn to two given intersecting circles.

179. If the three common chords of three circles which intersect each other, two and two, be produced, they will intersect in a point.

180. If three circles touch two and two, the tangents at the points of contact meet at a point and are equal.

181. The tangents at the points A, B on a circle whowe

centre is O intersect in D. Prove that the tangents at the extremities of all chords of the circle which are bisected by AB intersect upon the circle whose diameter is OD.

*182. The feet of the perpendiculars from any point on the circumference of the circumscribed circle of a triangle on the sides of the triangle lie in a straight line.

*183.

Describe a circle through two given points and touching a given straight line.

*184.

Describe a circle through two given points and touching a given circle.

185. A triangle has two of its angles each double the third. Prove that the circle through the middle points of its sides will intercept portions from the equal sides which are sides of a regular pentagon inscribed in the circle.

186. Construct a square such that two of its sides shall pass through two given points, and the remaining two intersect in another given point.

187. A quadrilateral can be inscribed in and circumscribed about a circle; prove that the straight lines joining the opposite points of contact of the inscribed circle are at right angles to each other.

If perpendiculars be drawn from the extremities of the diameter of a circle on any chord or any chord produced, the rectangle contained by the segments between the feet of the perpendiculars and either extremity of the chord is equal to that contained by the perpendiculars.

189. Two points D, E are taken in the base of a triangle ABC so that the tangents drawn from the points B, C to the circle circumscribing ADE are equal: shew that this requires that D and E shall be equidistant from B and C respectively.

190. The four sides of any quadrilateral inscribed in a circle being given, shew that the area does not depend upon the order in which the sides are placed in the circle. Will the order of the sides affect the magnitude of the angles of the figure?

191. The circumference of one circle passes through the centre of another circle. If from any point in the circumference of the former circle two straight lines be drawn to touch the latter circle, prove that the straight line joining the points of contact is bisected by the common chord of the circles.

192. If a regular pentagon ABCDE be inscribed in a circle and P be the middle point of the arc AB, prove that AP and the radius of the circle are together equal to PC.

193. When four circles can be described to touch three given straight lines the square on the distance between the centres of any two together with that on the distance between the centres of the other two is equal to the square on the diameter of the circle through the centres of any three.

194. If P be a point in a diameter of a circle and PT the perpendicular on the tangent at a point Q, then rectangle PT, AB is equal to the rectangle AP, PB together with the square on PQ.

DEFINITIONS OF BOOK III.

DEF. I. A circle is a plane figure contained by one line, which is called the circumference, and is such that all straight lines drawn from a certain point within the figure to the circumference are equal to one another. This point is called the centre of the circle. A radius of a circle is a straight line drawn from

DEF. 2.

the centre to the circumference.

DEF. 3.

Two arcs

A diameter of a circle is a straight line drawn through the centre, and terminated both ways by the circumference. DEF. 4. An aro is a part of a circumference. which together make the whole circumference are said to be conjugate. The greater of the two is called the major conjugate arc and the smaller the minor conjugate arc.

DEF. 5. The conjugate angles formed at the centre of a circle by two radii are said to stand upon the conjugate arc opposite them intercepted by the radii, the major angle upon the major arc, and the minor angle upon the minor arc.

DEF. 6. A sector is a figure contained by an arc and the

radii drawn to its extremities.

The angle of the sector is the angle

at the centre which stands upon the arc of the sector.

DEF. 7. A chord of a circle is the straight line joining any

two points on the circumference.