1851. Iv. 16. In a given circle inscribe a triangle, whose angles are as the numbers 2, 5 and 8.

1852.

1. 42. Divide a triangle by two straight lines into three parts, which, when properly arranged, shall form a parallelogram whose angles are of given magnitude.

11. 12. Triangles are described on the same base and having the difference of the squares on the

other sides constant: shew that the vertex of any triangle is in one or other of two fixed straight lines.

IV. 3. Two equilateral triangles are described about the same circle: shew that their intersections

will form a hexagon equilateral, but not generally equiangular.

1853. 1. B. Cor. If lines be drawn through the extremities of the base of an isosceles triangle, making angles

with it, on the side remote from the vertex, each equal to one third of one of the equal angles, and meeting the sides produced, prove that three of the triangles thus formed are isosceles.

1. 29. Through two given points draw two lines, forming with a line, given in position, an equilateral triangle.

II. 11. In the figure, if I be the point of division of the given line AB, and DA be the side of the square which is bisected in E and produced to F, and if DH be produced to meet BF in L, prove that DL is perpendicular to BF, and is divided by BE similarly to the given line. 111. 32. Through a given point without a circle draw a chord such that the difference of the angles

in the two segments, into which it divides the circle, may be equal to a given angle.

III. 36. From a given point as centre describe a circle cutting a given line in two points, so that the rectangle contained by their distances from a fixed point in the line may be equal to a given square

1854.

I. 43. If K be the common angular point of the parallelograms about the diameter, and BD the other diameter, the difference of the parallelograms is equal to twice the triangle BKD. II. 11. Produce a given straight line to a point such that the rectangle contained by the whole line thus produced and the part produced shall be equal to the square on the given straight line.

III. 22. If the opposite sides of the quadrilateral be produced to meet in P, Q, and about the triangles so formed without the quadrilateral circles be described meeting again in R, shew that P, R, Q will be in one straight line.

IV. 10. Upon a given straight line, as base, describe an isosceles triangle having the third angle

treble of each of the angles at the base.

1855. 1. 20. Prove that the sum of the distances of any point from the three angles of a triangle is greater than half the perimeter of the triangle.

I. 47. If a line be drawn parallel to the hypotenuse of a right-angled triangle, and each of the acute angles be joined with the points where this line intersects the sides respectively opposite to them, the squares on the joining lines are together equal to the squares on the hypotenuse and on the line drawn parallel to it.

II. 9. Divide a given straight line into two parts, such that the square on one of them may be double of the square on the other, without employing the Sixth Book.

111. 27. If any number of triangles, upon the same base BC, and on the same side of it, have their vertical angles equal, and perpendiculars meeting in D be drawn from B, C upon the opposite sides, find the locus of D, and shew that all the lines which bisect the angle BDC pass through the same point.

1855. iv. 4. If the circle inscribed in a triangle ABC touch the sides AB, AC in the points D, E, and a straight line be drawn from A to the centre of the circle, meeting the circumference in G, shew that G is the centre of the circle inscribed in the triangle ADE.

1856.

1. 34. Of all parallelograms, which can be formed with diameters of given length, the rhombus is

the greatest.

II. 12. If AB, one of the equal sides of an isosceles triangle ABC, be produced beyond the base to D, so that BD=AB, shew that the square on CD is equal to the square on AB together with twice the square on BC.

IV. 15. Shew how to derive the hexagon from an equilateral triangle inscribed in the circle, and from this construction shew that the side of the hexagon equals the radius of the circle, and that the hexagon is double of the triangle.

1857. 1. 35. ABC is an isosceles triangle, of which A is the vertex: AB, AC are bisected in D and E respectively; BE, CD intersect in F: shew that the triangle ADE is equal to three times the triangle DEF.

II. 13. The base of a triangle is given, and is bisected by the centre of a given circle, the circumference of which is the locus of the vertex :

prove that the sum of the squares on the two sides of the triangle is invariable.

III. 22. Prove that the sum of the angles in the four segments of the circle, exterior to the quadrilateral, is equal to six right angles.

IV. 4. Circles are inscribed in the two triangles formed by drawing a perpendicular from an angle of a triangle upon the opposite side, and analogous circles are described in relation to the two other like perpendiculars prove that the

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.

1861.

1. 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 ad-
jacent sides is given.

111. 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.

I. 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.

II. 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 Fintersect in a point on AB produced.