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A straight line moves so that equal chords are cut off from it by two unequal circles which touch externally; prove that the line drawn at right angles to this straight line, through` its intersection with the tangent at the common point, passes through the point midway between the centres.

4. In a given circle inscribe a regular hexagon.

If DA be one side of the hexagon, AB a tangent equal in length to AD, and making an obtuse angle with it, C the centre of the circle, and if BD meet the circle in E, and BC meet the nearer part of the circumference in F, prove that AE and EF are equal to sides of regular polygons, in the circle, of twelve and twenty-four sides respectively.

Two equal circles touch at A; a circle of twice the radius is described having internal contact with one of them at B, ana cutting the other in P, Q; prove that the straight line AB will pass through either P or Q.

Four straight lines in a plane form four finite triangles; prove that the centres of the circles circumscribing these tri angles lie on a circle which passes also through the common point of the circumscribing circles.

AB is a common tangent to two circles, CD their common. chord; prove that if tangents be drawn from A to any other circle through C, D, the chord of contact will pass through B

Fan. 1875.

1. Parallelograms and triangles upon the same base and between the same parallels are equal.

A, B, C are the middle points of the sides of the triangle

ABC, and through A, B, C are drawn three parallel straight lines meeting BC, C1A1, A1B1, in a b c respectively; prove that the triangle abc is half the triangle ABC, and that bc passes through A, ca through B, ab through C.

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

If the diagonals AC, BD of the quadrilateral ABCD, inscribed in a circle, the centre of which is at O, intersect at right angles, in a fixed point P, prove that the feet of the perpendiculars drawn from O and P to the sides of the quadrilateral lie on a fixed circle, the centre of which is at the middle point of OP.

3. Upon a given straight line describe a segment of a circle which shall contain an angle equal to a given rectilineal angle.

Through a fixed point O any straight line OPQ is drawn, cutting a fixed circle in P and Q, and upon OP and OQ as chords are described circles touching the fixed circle at P and Q; prove that the two circles so described will intersect on another fixed circle.

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

Prove that the circle drawn through the middle points of the sides of this triangle will intercept portions of the equal sides such that a regular pentagon can be inscribed in the circle having these portions as two of its sides.

Upon the sides of a triangle, ABC as bases, are described

three equilateral triangles, a BC, ьCA, and c AB, all upon the same side of their bases as the triangle ABC; prove that A a, Bb, Cc are all equal and pass through a point which lies on all the three circles, circumscribing the equilateral triangles..

Give the circumscribed and inscribed circles of a triangle; prove that the centres of the escribed circles lie on a fixed circle.

Fan. 1876.

1. The opposite sides and angles of a parallelogram are equal to one another.

Prove that, if O be any point in the plane of a parallelogram ABCD, and the parallelograms OAEB, OBFC, OCGD, ODHA be completed, then EFGH will be a parallelogram whose area is double that of the parallelogram ABCD.

2. In every triangle the square on a side subtending an acute angle is less than the squares on the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle and the acute angle.

By means of this proposition, show that the locus of a point, whose distances from two fixed points are in a constant ratio, is a circle.

3. If from any point without a circle two straight lines be drawn, one of which cuts the circle, and the other meets it, and if the rectangle contained by the whole line which cuts the circle and the part of it without the circle be equal

to the square on the line which meets the circle, the line which meets the circle will touch it.

Describe, through two given points, a circle such that the chord intercepted by it on a given unlimited straight line may be of given length.

Fan. 1877.

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

If through D, the middle point of the hypotenuse BC, the straight line DE be drawn at right angles to BC, meeting a side AC in E; prove (by the method of the First Book) that the rectangle contained by EC, AC is half the square on BC, and that the square on EC is equal to the squares on EA, AB.

2. In every triangle, the square on the side subtending an acute angle is less than the squares on the sides containing that angle by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall on it from the opposite angle and the acute angle.

Prove that if a circle be described with its centre on a fixed circle, and passing through a fixed point, the perpendicular from the fixed point on the common chord of the two circles will be of constant length.

3. If two straight lines cut one another within a circle, the rectangle contained by the segments of one of them is

equal to the rectangle contained by the segments of the other.

Prove that if through a fixed point O any straight line be drawn meeting a fixed circle, whose centre is C, in P and P1, the circle described round PCP1 will pass through a fixed point D in OC, and that if OC meet the fixed circle in A, AP will bisect the angle OPD.

4. Inscribe an equilateral and equiangular pentagon in a given circle.

If ABCDE be the pentagon, and if P be the middle point of the arc AB, prove that AP, together with the radius of the circle, is equal to PC.

Prove that, if the tangents at B and C to the circle ABC meet in O, the chord of the circle drawn through O parallel to AB will be bisected by AC.

Describe a square whose sides shall pass respectively through four given points.

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