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89. In the same case, the sum of the squares of the sides of the two new triangles is equal to the sum of the squares of the sides of the original triangle.

90. If R, r denote the radii of the circumscribed and inscribed circles to a regular polygon of any number of sides R', r', corresponding radii to a regular polygon of the same area, and double the number of sides, prove

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91. If the altitude of a triangle be equal to its base, the sum of the distances of the orthocentre from the base and from the middle point of the base is equal to half the base.

92. In any triangle, the radius of the circumscribed circle is to the radius of the circle which is the locus of the vertex, when the base and the ratio of the sides are given, as the difference of the squares of the sides is to four times the area.

93. Given the area of a parallelogram, one of its angles, and the difference between its diagonals; construct the parallelogram.

94. If a variable circle touch two equal circles, one internally and the other externally, and perpendiculars be let fall from its centre on the transverse tangents to these circles, the rectangle of the intercepts between the feet of these perpendiculars and the intersection of the tangents is constant.

95. Given the base of a triangle, the vertical angle, and the point in the base whose distance from the vertex is equal half the sum of the sides; construct the triangle.

96. If the middle point of the base BC of an isosceles triangle ABC be the centre of a circle touching the equal sides, prove that any variable tangent to the circle will cut the sides in points D, E such that the rectangle BD. CE will be constant.

97. Inscribe in a given circle a trapezium, the sum of whose opposite parallel sides is given, and whose area is given.

98. Inscribe in a given circle a polygon all whose sides pass through given points.

99. If two circles X, Y be so related that a triangle may be inscribed in X and circumscribed about Y, an infinite number of such triangles can be constructed

100. In the same case, the circle inscribed in the triangle formed by joining the points of contact on Y touches a given circle.

101. And the circle described about the triangle formed by drawing tangents to X at the angular points of the inscribed triangle touches a given circle.

102. Find a point, the sum of whose distances from three given points may be a minimum.

103. A line drawn through the intersection of two tangents to a circle is divided harmonically by the circle and the chord of

contact.

104. To construct a quadrilateral similar to a given one whose four sides shall pass through four given points.

105. To construct a quadrilateral, similar to a given one, whose four vertices shall lie on four given lines.

106. Given the base of a triangle, the difference of the base angles, and the rectangle of the sides; construct the triangle.

107. ABCD is a square, the side CD is bisected in E, and the line EF drawn, making the angle AEF = EAB; prove that EF divides the side BC in the ratio of 2: 1.

108. If any chord be drawn through a fixed point on a diameter of a circle, and its extremities joined to either end of the diameter, the joining lines cut off, on the tangent at the other end, portions whose rectangle is constant.

109. If two circles touch, and through their point of contact two secants be drawn at right angles to each other, cutting the circles respectively in the points A, A'; B, B'; then AA'2 + BB'2 is constant.

110. If two secants at right angles to each other, passing through one of the points of intersection of two circles, cut the circles again, and the line through their centres in the two systems of points a, b, c; a, b, c respectively, then ab: bc:: a'b': b'c'.

111. Two circles described to touch an ordinate of a semicircle, the semicircle itself, and the semicircles on the segments of the diameter, are equal to one another.

112. If a chord of a given circle subtend a right angle at a given point, the locus of the intersection of the tangents at its extremities is a circle.

113. The rectangle contained by the segments of the base of a triangle, made by the point of contact of the inscribed circle, is equal to the rectangle contained by the perpendiculars from the extremities of the base on the bisector of the vertical angle.

114. If O be the centre of the inscribed circle of the triangle ABC, prove

OA2 OB2 OC2

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115. State and prove the corresponding theorems for the centres of the escribed circles.

116. Four points A, B, C, D are collinear; find a point P at which the segments AB, BC, CD subtend equal angles.

117. The product of the bisectors of the three angles of a triangle whose sides are a, b, c is

8 abc.s.area

(a+b) (b + c) (c + a)*

118. In the same case the product of the alternate segments of the sides made by the bisectors of the angles is

a2b2c2

(a+b) (b+c) (e+a)'

119. If three of the six points in which a circle meets the sides of any triangle be such that the lines joining them to the opposite vertices are concurrent, the same property is true of the three remaining points.

120. If a triangle A'B'C' be inscribed in another ABC, prove

AB'. BC'. CA' + A'B. B'C. C'A.

is equal twice the triangle A'B'C' multiplied by the diameter of the circle ABC.

121. Construct a polygon of an odd number of sides, being given the sides taken in order are divided in given ratios by fixed points.

122. If the external diagonal of a quadrilateral inscribed in a given circle be a chord of another given circle, the locus of its middle point is a circle.

123. If a chord of one circle be a tangent to another, the line connecting the middle point of each arc which it cuts off on the first to its point of contact with the second passes through a given point.

124. From a point P in the plane of a given polygon perpendiculars are let fall on its sides; if the area of the polygon formed by joining the feet of the perpendiculars be given, the locus of P is a circle.

BOOK XI.

THEORY OF PLANES, COPLANAR LINES, AND SOLID

ANGLES.

DEFINITIONS.

I. When two or more lines are in one plane they are said to be coplanar.

II. The angle which one plane makes with another is called a dihedral angle.

III. A solid angle is that which is made by more than two plane angles, in different planes, meeting in a point.

IV. The point is called the vertex of the solid angle. v. If a solid angle be composed of three plane angles it is called a trihedral angle; if of four, a tetrahedral; and if of more than four, a polyhedral angle.

PROP. I.-THEOREM.

One part (AB) of a right line cannot be in a plane (X) and another part (BC) not in it.

X

Dem. Since AB is in the plane X, it can be produced in it [Bk. 1. Post. II.]; let it be produced to D. Then, if BC be not in X, let any other plane passing through AD be turned round AD until it passes through the point C. Now,

A

B

D

because the points B, C are in this second plane, the line

BC [1., Def. vI.] is in it. Therefore the two right lines ABC, ABD lying in one plane have a common segment AB, which is impossible. Therefore, &c.

PROP. II.-THEOREM.

Two right lines (AB, CD) which intersect one another in any point (E) are coplanar, and so also are any three right lines (EC, CB, BE) which form a triangle.

D

Dem. Let any plane pass through EB and be turned round it until it passes through C. Then because the points E, C are in this plane, the right line EC is in it [1., Def. vI.]. For the same reason the line BC is in it. Therefore the lines EC, CB, BE are coplanar; but AB and CD are two of these lines. Hence AB and CD are coplanar. c4

PROP. III.-THEOREM.

B

If two planes (AB, BC) cut one another, their common section (BD) is a right line.

Dem.-If not from B to D, draw in the plane AB the

right line BED, and in the

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If a right line (EF) be perpendicular to each of two intersecting lines (AB, CD), it will be perpendicular to any line GH, which is both coplanar and concurrent with them.

Dem.-Through any point G in GH draw a line BC

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