sum of the diameters of the six circles together with the sum of the sides of the original triangle is equal to twice the sum of the three perpendiculars. 1858. 1. 28. Assuming as an axiom that two straight lines cannot both be parallel to the same straight line, deduce Euclid's sixth postulate as a corollary of the proposition referred to. II. 7. Produce a given straight line, so that the sum of the squares on the given line and the part produced may be equal to twice the rectangle contained by the whole line thus produced and the produced part. III. 19. Describe a circle, which shall touch a given straight line at a given point and bisect the circumference of a given circle. 1859. I. 41. Trisect a parallelogram by straight lines drawn from one of its angular points. II. 13. Prove that, in any quadrilateral, the squares on the diagonals are together equal to four times the sum of the squares on the straight lines joining the middle points of opposite sides. III. 31. Two equal circles touch each other externally, and through the point of contact chords are drawn, one to each circle, at right angles to each other: prove that the straight line, joining the other extremities of these chords, is equal and parallel to the straight line joining the centres of the circles. iv. 4. Triangles are constructed on the same base with equal vertical angles: prove that the locus of the centres of the escribed circles, each of which touches one of the sides externally and the other side and base produced, is an arc of a circle, the centre of which is on the circumference of the circle circumscribing the triangles. 1860. L. 35. If a straight line DME be drawn through the middle point M of the base BC of a triangle ABC, so as to cut off equal parts AD, AE from the sides AB, AC, produced if necessary, respectively, then shall BD be equal to CE. II. 14. Shew how to construct a rectangle which shall be equal to a given square; (1) when the sum, and (2) when the difference of two adjacent sides is given. III. 36. If two chords AB, AC be drawn from any point A of a circle, and be produced to D and E, so that the rectangle AC, AE is equal to the rectangle AB, AD, then, if O be the centre of the circle, AO is perpendicular to DE. IV. 10. If A be the vertex, and BD the base of the constructed triangle, D being one of the points of intersection of the two circles employed in the construction, and E the other, and AE be drawn meeting BD produced in F, prove that FAB is another isosceles triangle of the same kind. 1861. L. 32. If ABC be a triangle, in which C is a right angle, shew how, by means of Book I., to draw a straight line parallel to a given straight line so as to be terminated by CA and CB and bisected by AB. 11. 13. If ABC be a triangle, in which C is a right angle, and DE be drawn from a point D in AC at right angles to AB, prove, without using Book III., that the rectangles AB, AE and AC, AD will be equal. III. 32. Two circles intersect in A and B, and CBD is drawn perpendicular to AB to meet the circles in C and D; if EAF bisect either the interior or exterior angle between CA and DA, prove that the tangents to the circles at E and F intersect in a point on AB produced. 1861. IV. 4. Describe a circle touching the side BC of the triangle ABC, and the other two sides produced, and prove that the distance between the points of contact of the side BC with the inscribed circle, and the latter circle, is equal to the difference between the sides AB and AC. 1862. 1. 4. Upon the sides AB, BC, and CD of a parallelogram ABCD, three equilateral triangles are described, that on BC towards the same parts as the parallelogram, and those on AB, CD towards the opposite parts. Prove that the distances of the vertices of the triangles on AB, CD, from that on BC, are respectively equal to the two diagonals of the parallelo 1863. gram. III. 10. Divide a given straight line into two parts, so that the squares on the whole line and on one of the parts may be together double of the square on the other part. III. 28. A triangle is turned about its vertex, until one of the sides intersecting in that vertex is in the same straight line as the other previously was prove that the line, joining the vertex with the point of intersection of the two positions of the base, produced if necessary, bisects the angle between these two positions. IV. 10. Prove that the smaller of the two circles, employed in Euclid's construction, is equal to the circle described about the required triangle. I. 47. Two triangles ABC, A'B'C' have their sides respectively parallel. BB1, CC1 are drawn perpendicular to B'C'; CC, AA, to CA'; and AA,, BB, to A'B'. Prove that the sum of the squares on AB1, BC2, CA, together, is equal to the sum of those on AC1, BA2, CB3 together. II. 11. Divide a given straight line into two parts, such that the rectangle contained by the whole and one part may be equal to that contained by the other part and a given straight line. 1863. III. 28. Two equal circles intersect in A, B; PQT perpendicular to AB meets it in T, and the circles in P, Q. AP, BQ meet in R; AQ, BP in S: prove than the angle RTS is bi sected by TP. 1864. 1. 38. If a quadrilateral figure have two sides parallel, and the parallel sides be bisected, the lines joining the points of bisection shall pass through the point in which the diagonals cut one another. IL. 14. Divide a given straight line (when possible) into three parts such that the rectangle contained by two of them shall be equal to a given rectilineal figure, and that the squares on these two parts shall together be equal to the square on the third. III. 36. If from a given point A without a given circle any two straight lines APQ, ARS, be drawn, making equal angles with the diameter which passes through A, and cutting the circle in P, Q, and R, S, respectively, then PS, QR, shall cut one another in a given point. Iv. 11. If a figure of any odd number of sides have all its angular points on the same circle, and all its angles equal, then shall its sides be equal. 1865. L. 20. Give a geometrical construction for finding a point in a given straight line, the difference of the distances of which from two given points on the same side of the line shall be the greatest possible. II. 12. The base BC of an isosceles triangle ABC is produced to a point D; AD is joined, and in AD a point E is taken, such that the rectangle AD, AE, is equal to the square on either of the equal sides AB, AC, of the triangle: prove that the rectangle BD, CD is equal to the rectangle AD, ED. 1865. III. 18. A given straight line is drawn at right angles to the straight line joining the centres of two given circles: prove that the difference between the squares on two tangents drawn, one to each circle, from any point on the given straight line, is constant. 1866. iv. 5. Having given one side of a triangle, and the centre of the circumscribed circle, determine the locus of the centre of the inscribed circle. 1. 33. Prove that a quadrilateral, which has two opposite sides and two opposite obtuse angles equal, is a parallelogram. Shew that the figure is not necessarily a parallelogram, if the equal angles are acute. II. 9. Prove this also by superposition of the squares or their halves. III. 22. If four circles be drawn, each passing through three out of four given points, the angle between the tangents at the intersection of two of the circles is equal to the angle between the tangents at the intersection of the other two circles. IV. 2. In a given circle inscribe a triangle such that two of the sides of the triangle shall pass through given points and the third side be at a given distance from the centre of the given circle. 1867. 1. 16. Any two exterior angles of a triangle are together greater than two right angles. L 43. What is the greatest value which these complements, for a given parallelogram, can have? I. 11. Divide a given straight line into two parts such that the squares on the whole line and on one of the parts shall be together double of the square on the other part. |