lateral is greater than the sum of the squares on the straight lines joining F to the same points by four times the square on EF

3. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles 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.

AB, CD are parallel diameters of two cir.les, and AC cuts the circles in PQ: prove that the tangents to the circles at PQ are parallel.

4. Describe an isosceles triangle, having each of the angles at the base double of the third angle.

Hence show how to describe an equilateral and equiangular pentagon about a circle without first inscribing one.

AB is a fixed chord of a circle PQ, any other chord which is bisected by AB; prove that the tangents at P and Q meet on a fixed circle.

In a pentagon five straight lines can be drawn by joining each angular point to the next angular point but one; if four of these straight lines are parallel to the opposite sides of the pentagon and the perpendicular distances between these parallels are equal, prove that the pentagon is regular.

A and B are two fixed points on a circle from which tangents are drawn to any concentric circle; the chords joining the points of contact which are not parallel to AB will pass through a fixed point.

Fan. 1871.

1. When is one proposition said to be converse to another?

State and prove the proposition or propositions converse to the following:- Triangles on equal bases and between the same parallels are equal to each other.'

Through the angular points A, B, C of a triangle are drawn three parallel straight lines meeting the opposite sides in A, B, C1 respectively; prove that the triangles AB1C1, BC11, CA1B1 are all equal.

2. If a straight line be bisected, and produced to any point, the square on the whole line thus produced, and the square on the part of it produced, are together double of the square on half the line bisected and of the square on the line made up of the half and the part produced.

Produce a given straight line so that the square on the whole line thus produced may be double the square on the part produced.

3. If a straight line touch a circle, and from the point of contact a straight line be drawn cutting the circle, the angles 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.

The opposite sides of a quadrilateral inscribed in a circle are produced to meet in P, Q, and about the four triangles thus formed circles are described; prove that the tangents to these circles at P and Q form a quadrilateral equal in all respects to the original, and that the line joining the centres of the circles about the two quadrilaterals bisects PQ.

4. Describe a circle about a given triangle.

A triangle is inscribed in a given circle so as to have its centre of perpendicular at a given point; prove that the middle points of its sides lie on a fixed circle.

A quadrilateral is divided into four equal triangles by lines joining its angular points to a point within it, prove that one of its diagonals must be bisected by the other.

Fan. 1872.

1. In any right-angled triangle, the square described on the side subtending the right angle is equal to the sum of the squares described on the sides which contain the right angle.

If CE, BD be the squares described upon the side AC and the hypotenuse AB, and if EB, CD intersect in F, prove that AF bisects the angle EFD.

2. Describe a square equal to a given rectilineal figure.

If the given rectilineal figure be that of Euclid I. 47, show how to determine the required square graphically.

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

Two circles intersect in A, B: PAP1, QAQ1 are drawn equally inclined to AB to meet the circles in P, P1, Q, Q1 ; prove that PP is equal to QQ1.

4. Inscribe a circle in a given triangle.

Having given an angular point of a triangle, the circumscribed circle, and the centre of the inscribed circle, construct the triangle.

Fan. 1873.

1. If two triangles have two angles of the one equal to two angles of the other, each to each, and one side equal to one side, namely, the sides opposite to equal angles in each, the other sides shall be equal, each to each, and the third angle of the one equal to the third angle of the other.

Prove that the point of intersection of the diagonals of a square, described on the hypotenuse of a right-angled triangle, is equidistant from the two sides containing the right angle.

2. Equal triangles on the same base, and on the same side of it, are between the same parallels.

The angular points of one triangle lie on the sides of another; if the latter triangle be thus divided into four equal parts, prove that the lines joining its angular points with the corresponding angular points of the former triangle will be bisected by the sides of the former.

3. The angles in the same segment of a circle are equal to one another.

The point, in which the external bisector of one angle of a triangle again cuts the circumscribed circle, is equidistant from the other two angular points of the triangle and from the centres of two of the escribed circles.

4. Describe an isosceles triangle, having each of the angles at the base double of the third angle.

What portion of the circumference of a small circle does the large circle intercept?

Four circles are described, each passing through two adjacent angular points of a square, and also through a point P on one of the diagonals. A quadrilateral is described such that each angular point lies on the circumference of one of the circles, and each side passes through one of the angular points of the square. Prove that the quadrilateral may have a circle described about it with its centre at P, and that its diagonals are equal and at right angles.

Fan. 1874.

1. The opposite sides and angles of a parallelogram are equal, and the diagonal bisects it.

The bisectors of the angles of a parallelogram form a rectangle whose diagonals are parallel to the sides of the parallelogram, and equal to the difference between them.

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

Two triangles are formed by lines drawn through the points A, B, C perpendicular to the lines AB, BC, CA respectively in one case, and perpendicular to AC, BA, CB in the other case; prove that these two triangles are in all respects equal.

1

3. If from an external point two straight lines be drawn, one of which cuts a 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, will be equal to the square on the line which touches it.