IOI. From A, one of the points of intersection of two circles, two lines AXY, APQ are drawn at right angles; if X, Q are on one circle, and Y, P on the other, then

PQ2+ XY2..

=

NOTE-Use ii. Addenda (12).

4 (sq. on join of their centres).

102. If the circle through B, C and the orthocentre of a triangle ABC, meets the median from A (produced) in X, then AX is twice that median.

103. If the sides and angles of a triangle ABC, are given, and its position varies, subject only to the condition that AB, AC each go through a fixed point; then BC always touches a fixed circle.

Λ

NOTE—Let P be the fixed pt. in AB, and Q in AC; on PQ describe a segt. contg. an A; and draw PO to cut off segt. contg. an B: O is cent. of the required O.

104. If four circles are drawn, each passing through three out of four fixed points; then one of the angles between the tangents at the intersection of one pair of circles is equal to one of the angles between the tangents at the intersection of the other pair.

105. If two Simson's Lines are drawn with respect to points at the ends of a diameter, then

1o, the lines are at right angles;

2o, they intersect on the N. P. circle.

NOTE-If PCQ is diam., PM, PL, QX, QY 1s on sides aẞ, ay; then 1o, comes from considering λs of cyclic quads. PMaL, QXaY; and 2o, comes from the facts that N is mid pt. CO, and that PO, QO are bisected by the two Simson's Lines: cf. iii. Addenda (18) and (20).

106. If ABC is a triangle, and P any point, the N. P. circles of the triangles PAB, PBC, PCA, intersect in a point X, on the N. P. circle of ABC. If P is on the circum-circle of ABC, then

1o, the Simson's Lines of the triangles PAB, PBC, PCA, ABC, rela tively to C, A, B, P, respectively, are concurrent in X; and

2o, the centroids of the same triangles are concyclic. (Prof. Bordage: Educational Times: Reprint; Vol. XLV.)

NOTE-The first part of the Theorem may be put thus—

The four triangles determined by any four points, taken three and three together, are such that their N. P. circles have a common point.

Take the N. P. O as circumscribing the ▲s formed by the joins of mid pts.; and use the converse of iii. 22.

EXAMPLES OF Loci.

NOTE-In each of the following the Locus is to be determined as completely as possible: if the Locus is a straight line, its position with regard to given points or lines-and if a circle, its centre and radius—are to be found.

107. Four rods are pivoted together, so that the pivots are the corners of a rhombus, and the framework is capable of motion in one plane; if one rod is fixed, and the others are moved about, then the locus of the intersection of the diagonal lines joining the pivots is a circle.

108. The ends of a rod of given length slide on fixed straight intersecting wires; if perpendiculars to the wires at, or from, the ends of the rod are drawn, the loci of their intersection are two circles concentric at the intersection of the wires.

109. The locus of the mid point of all lines drawn from a fixed point to meet a fixed circle, is a circle.

NOTE-This is a particular case of a useful Theorem given hereafter: vi. Addenda (11).

110. If in a fixed circle a triangle is inscribed, so as to have its orthocentre at a fixed point; then the locus of the mid points of its sides is a circle.

NOTE-Produce join of orthocent. and one mid. pt. a dist. equal to itself; show that the extremity of this lies on the ; and then use the preceding Exercise.

III. If the extremities of a diameter of a circle are joined transversely to the extremities of a chord of constant length, the locus of the intersection of the joins is a circle.

112. If a chord of a given circle subtends a right angle at a given point (within or without the circle) the locus of its mid point is a circle, whose centre is the mid point of the join of the centre of the given circle to the fixed point.

NOTE-Use Locus (n) p. 178.

113. If a quadrilateral circumscribes a circle, the join of the mid points of its diagonals goes through the centre of the circle.

NOTE-This is a particular case of Newton's Principia, Lib. I, Lemma 25, Cor. (3); and may be easily done by using Locus on p. 179.

114. Given the base and vertical angle of a triangle, find the Locus of its— 1o, in-centre: 2°, ortho-centre: 3o, centroid: 4°, ex-centres: 5o, N.P. centre. NOTE-5° can be made to depend on Ex. 109 above.

115. Two variable circles, each touching the same given line at a given point (different for each circle) also touch each other: find the Locus of the point of contact of the circles.

116. Two opposite corners of a given square move on two lines at right angles: find the Loci of the other corners.

117. Given the base of a triangle in magnitude and position; and given also, 1o, the sum of the other two sides, or 2o, their difference; find the Loci of the feet of perpendiculars from the ends of the base on the bisector of the external vertical angle for the sum, and vertical angle for the difference.

118. Given an angle of a triangle, in position and magnitude, and given also the sum of the sides which contain it; find the Locus of its circum-centre. NOTE-Use iii. Addenda (15).

119. Given a circle, and a fixed point within it; find the Locus of the intersection of tangents at the extremities of all chords through the point. 120. Given two fixed circles, 1°, intersecting, 2o, not intersecting; find, in each case, the Locus of the points from which tangents to the circles are equal. 121. Given two fixed points, find the Locus of a variable point whose distance from one of the points is twice its distance from the other.

NOTE-This is a particular case of an important Locus, given on p. 294. 122. AB is a fixed chord of a fixed circle, AP a variable chord of the same circle; find the Locus of the mid point of BP.

123. Through a point P within a rectangle ABCD, parallels are drawn to the sides; and P moves so that the difference of the rectangles PA, PC is constant; find the Locus of P.

124. The vertex of an isosceles triangle (given in all but position) moves along the circumference of a fixed circle (whose radius is equal to one of the equal sides) and one of the extremities of the base moves along a fixed diameter of the circle; find the Locus of the other extremity

125. A line of given length moves with its extremities on two fixed intersecting lines; find the Locus of the orthocentre of the triangle thus formed.

126. A point, inside or outside a circle, is joined to two fixed points on its circumference; if the joins intercept a constant arc, find the Locus of the point. 127. AB is a fixed line, O a fixed point, and MN a fixed length; if a variable point P moves so that (PQ being the perpendicular from it on AB)

NOTE-Draw OXL to AB, and produce XO to C, so that OC

then it can be shown that CP is constant, and . that C is the centre of the which is Locus of P. Cf. p. 364.

BOOK iv.

Proposition 1.

PROBLEM-In a given circle to draw (when possible) a chord equal to a given straight line.

Since no chd. of a O can be greater than its diam., the prob. will not be possible unless the given st. line is not greater than the diam.

Take any pt. C on the circumf. of given O.
With centre C, and given line as radius, describe a O.
Let P be one of the pts. in which the Os cut.

Join CP.

Then CP given line, and is a chd. of given O.

NOTE-A line drawn (as above) from a given point, to meet a given circle (or line) and be of a given length, is said to be inflected from the point to the circle (or line).

Def. When all the corners of one rectilineal figure are on the sides of another rectilineal figure, the

first figure is said to be inscribed in

the second; and the second figure is

said to be circumscribed about the first.

Def. When all the corners of a rectilineal figure are on the circumference of a circle, the rectilineal

figure is said to be inscribed in the circle; and the circle is said to be circumscribed about the rectilineal

figure.

Def. When each side of a rectilineal figure touches a circle,

the rectilineal figure is said to be circumscribed about the circle; and the circle is said to be inscribed in the rectilineal figure.

Def. When a rectilineal figure is equiangular and equilateral it is called regular.

Def. If the angles of a rectilineal figure (taken successively) are equal respectively to those of another (also taken successively) then the figures are said to be equiangular to each other; each angle of the one is said to correspond to the angle equal to it in the other; and the sides joining the vertices of corresponding angles are called corresponding sides.