and the three interior angles of every triangle are together equal to two right angles.

Construct a right angled isosceles triangle of given perimeter.

2. If the square described on one of the sides of a triangle be equal to the squares described on the other two sides of it, the angles contained by these two sides is a right angle.

If a square be described on the diagonal of a square and another square on the diagonal of this, and so on, etc., the last square so described will be equal to the sum of all the others so described, with twice the original square.

3. To divide a given straight line into two parts so that the rectangle contained by the whole and one of the parts may be equal to the square on the other part.

If the constructed figure be completed to form an oblong, the diagonals of the oblong will not both pass through the point of division of the given line.

Section II.

1. If a straight line drawn through the centre of a circle cut a straight line, which does not pass through the centre at right angles, it shall besect it.

A circle passes through three of the angular points of a rhombus; show that it cannot pass through the fourth. 2. Equal straight lines in a circle are equally distant from the centre.

A circle is described through the angular points of an oblong, whose diameter is twice as long as the shorter side, show that the shorter sides are sides of a regular hexagon inscribed in the circle.

3. To draw a straight line from a given point either without or within the circumference, which shall touch a given circle.

Find the locus of the centres of all circles touching two equal circles.

Section III.

1. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If any six sided figure be inscribed in a circle, the sum of either three alternate angles is equal to four right angles.

2. In equal circles equal angles stand on equal arcs, whether they be at the centre or circumference.

Divide the circumference of a circle into three parts, which shall be in the ratios of 3 4 5.

3. If a straight line touch a circle and from the point of contact a straight line be drawn cutting the circle, the angle which this line makes with the line touching the circle shall be equal to the angles which are in the alternate segments of the circle.

If a polygon of any number of sides be inscribed in a circle, one side being a diameter of the circle, the sum of the squares on all its sides will be less than twice the square circumscribed about the circle.

Section IV.

1. If from any point without a circle two straight lines be drawn, one of which not passing through the centre cuts the circle and the other touches it, the rectangle contained by the whole line which cuts the circle and the part of it without the circle shall be equal to the square on the line which touches it.

If any number of circles intersect two and two so that their common chords pass through one point, a circle can be described through all the points in which tangents from this point meet the circles.

2. In a given circle to inscribe a triangle equiangular to a given triangle.

An equilateral triangle is inscribed in a circle and tangents are drawn through its angular points; show that another equilateral triangle will be formed four times as large as the original triangle

3. To describe a circle about a given triangle.

Four circles are described through the angular points of a rhombus, taken by threes; another rhombus will be formed by joining the centres of these circles.

Section V.

1. To inscribe a square in a given circle.

If lines be drawn from any point in the circumference of the circle to the angular points of the square, the squares on the first and third are equal to the squares on the second and fourth.

2. To inscribe an equilateral and equiangular pentagon in a given circle.

If the sides of the pentagon be produced to meet, another regular pentagon will be formed by joining the points of intersection.

3. The straight lines which bisect the angles of a triangle meet at the same point.

Section VI.

1. AB is bisected in C, and circles which do not intersect are described round A and B, two of their common tangents will also touch a third circle described round C, if the radius of this last circle be equal to half the sum of the two other radii.

2. Find the locus of the point outside a given circle, from which two tangents can be drawn to form an

equilateral triangle with the chord passing through the points of contact.

3. DE, FG, are chords of a circle parallel to a diameter AB and each at a distance from AB equal to one fourth of AB; AO, AR are chords bisected by DE, FG respectively; show that the quadrilateral figure AORR is equal to the square on AO.

APPENDIX D.

QUESTIONS

SET AT THE

Science and Art Department Examinations.

1875-77.

SUBJECT V.-PURE MATHEMATICS.

GEOMETRY.

Stage I.

May 1875.

1. Prove that the angles at the base of an isosceles triangle are equal to one another.

2. Prove that any two sides of a triangle are together greater than the third.

3. Let ABC be three points in a straight line, taken in order, and D any other point not in that line. Join B D. Show that the lines which bisect the angles ABD and DBC are perpendicular to one another.

4. Show that a line drawn from the angle (or vertex) of a triangle, so as to bisect the base, also bisects the triangle. 5. Prove that the diagonals of a parallelogram bisect one another.

6. Show that if a straight line meets two parallel straight lines, the alternate angles are equal. Write out at full length what definition of parallelism, and what axiom (if