and making the angle BCF equal to the angle ABE. Prove that AE is to EC as BF is to BD.

75. Find the locus of a point such that the tangents from it to each of two given circles may contain equal angles.

76. If a quadrilateral ABCD, having the sides BC, CD equal be inscribed in a circle, the rectangle AB, AD, together with the square on BC, will be equal to the square on AC.

77. On a level plain are to be seen two church spires; a man walks on the plain so that he always sees the spires at equal angles of elevation. Prove that he walks in a circle.

78. ABC and ABF are triangles on the same base in the ratio of 2 to 1; AF and BF produced meet BC and AC in D and E; in FB, FG is cut off equal to FE, and BG is bisected in O. Prove that BO: BE:: DF: DA.

79. From each of two opposite angular points of a parallelogram lines are drawn to the points of bisection of the sides containing the opposite angle. Shew that these lines will form

a parallelogram whose area will be one-third of that of the whole parallelogram.

80. On a given straight line construct a right-angled triangle whose three sides shall be in continued proportion.

81. ABC is an isosceles triangle whose vertex is A. From BA, CA cut off BD, CE, each equal to BC, and join BE, CD, cutting each other in F. Shew that if AC is equal to 25 BC, it is also equal to 35 FC.

82. Describe a triangle on a given base, and with a given vertical angle, such that the base may be a mean proportional between the sides. Shew that the problem is impossible, if the given angle is greater than the angle of an equilateral triangle.

DEFINITIONS OF BOOK V.

DEF. I. If the angles of a rectilineal figure, taken in order, are equal respectively to those of another, also taken in order, the figures are said to be equiangular. 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 termed corresponding sides.

DEF. 2. Similar figures are such as are equiangular, and have their sides proportional, the corresponding sides being homologous.

DEF. 3. Similar figures are said to be similarly situated upon given straight lines, when those straight lines are corresponding sides of the figures.

DEF. 4. The point of intersection of all straight lines which join the corresponding points of two similar figures, whose corresponding sides are parallel, is called the centre of similarity of the two figures.

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