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section of its diagonals AC and BD. If X is any point on AD and & B, XC, be joined, show that the area of the figure BXCO is one quarter that of the parallelogram.
7. Prove that the square on any straight line drawn from the vertex of an isosceles triangle to a point in the base is less than the square on a side of the triangle by the rectangle contained by the segments of the base.
9. Two circles touch one another externally, the diameter of one being double that of the other, and through the point of contact any straight line is drawn to meet the circumferences of both again; show that the part of the straight line cut off by the larger circle is double that cut off by the smaller.
10. Show that all chords of the same length in a circle will touch another circle with the same centre.
comble that estraight linces of both straight
B. Further Examination in Mathematics.
1. SEPTEMBER, 1902. 9. Describe an isosceles triangle having each of the angles at the base two and a half times the vertical angle.
10. Show that the sum of the squares on the diagonals of a parallelogram is equal to the sum of the squares on the sides.
11. If two circles cut one another, the straight line joining their centres bisects their common chord at right angles.
II. MICHAELMAS TERM, 1902. 2. Prove that the two sides of a triangle are together greater than twice the straight line drawn from the vertex to the middle point of the base.
4. AB is the hypotenuse of a right-angled triangle ABC. Find a point P in AB such that PB may be equal to the perpendicular from P on AC.
8. Two circles touch one another externally at the point X, and a straight line is drawn touching the circles at the points C and D. Show that CXD is a right angle. III. Hilary TERM, 1903. 7. If two equal triangles stand on opposite sides of the same base, the line joining their vertices is bisected by the base.
8. The straight lines drawn perpendicular to the sides of a triangle through their middle points meet in a point.
9. If two tangents be drawn to a circle from a point without it, the centre of the circle lies on the straight line bisecting the angle between them.
IV. SEPTEMBER, 1903. 4. D and E are the middle points of the sides BC, CA of the triangle ABC. AD and BE are joined and produced to A, B', so that DA'=AD and EB'=BE. Prove that A'CB' is a straight line, and is parallel to AB.
7. If D, E are the feet of the perpendiculars from the vertices A, B of the triangle A BC to the opposite sides, prove that a circle can be drawn through the points A, B, D, E.
V. MICHAELMAS TERM, 1903. 5. If A BC be a triangle, and D, E, F the middle points of its sides, prove that the area of the triangle DEF is equal to a quarter of the area of the triangle A BC.
8. If D be the middle point of the side BC of the triangle ABC, prove that the sum of the squares on AB, AC is equal to twice the sum of the squares on AD, BD.
VI. HILARY TERM, 1904. 3. ABCD is a parallelogram; E is a point on AB. Describe on AE as base a parallelogram equal to ABCD and having the angle EAD for one of its angles.
5. If a straight line be divided as in Euclid II. 11, prove that the sum of the squares on the whole line and on its smaller part is equal to thrice the rectangle contained by the whole and that part.
7. ABC, APQ are two straight lines. A circle which has P as centre passes through A and B. Another circle which has Q as centre passes through A and C. Prove that BP is parallel to CQ.
VII. TRINITY TERM, 1904. 3. The sides AB, AC of a triangle ABC are produced to D, E respectively, and the angles CBD, BCE are bisected by two straight lines meeting at F. Prove that AF bisects the angle BAC.
4. (6) Find the locus of a point which moves so that the sum of the squares of its distances from two fixed points is a constant area.
7. A, B, C, D are the vertices taken in order of a quadrilateral formed by four tangents to a circle. Prove that AB+CD= AD+ BC.
9. Prove that if A, B are two points situated on the same side of a straight line CD, two circles can, in general, be drawn to pass through A, B and to touch CD.
Examine the case where AB is parallel to CD.
VIII. SEPTEMBER, 1904. 4. A, B, C, D are the vertices taken in order of a quadrilateral whose diagonals are at right angles. Prove that AB? + CD2 = ADP + BC?. . 5. Prove that the straight line joining the iniddle points of two sides of a triangle divides the area of the
angle in the ratio 1. triangle divides the one iniddle
9. On a given circle four points A, B, C, D are taken, so that the angles DAB, DAC are equal. Prove that the straight line drawn from D at right angles to BC is a diameter of the circle.
10. Through the vertices A, B of a triangle ABC, AD, BE are drawn perpendicular respectively to the sides BC, CA and intersecting one another at F. Prove that CF is perpendicular to AB.
I. MICHAELMAS TERM, 1901. L. Quinctius Cincinnatus tilled his own little farm beyond the Tiber. The deputies of the Senate came thither early in the morning and found him digging in his field. And when he had sent to fetch his toga, and was now in fit guise to hear the message of the Senate, they hailed him dictator, and told him in what peril the consul and his army lay. So he went with four-andtwenty lictors before him to his house in Rome and chose L. Tarquitius, a brave man but poor, to be master of the horse. On that day the dictator made all business to cease in the Forum and summoned all who could bear arms to meet in the Field of Mars before sunset, ordering each man to bring with him victuals for five days and twelve wooden stakes. So at nightfall when everything was in readiness, the dictator marched with all speed to Mount Algidus. There he discovered the enemy's position, and the soldiers began digging a trench, driving in their stakes right round the Aequian camp.
II. HILARY TERM, 1902. When Cyrus received these tidings he turned to Croesus and said, 'Where will all this end, Čroesus, thinkest thou? It seemeth that these Lydians will not cease to cause trouble both to themselves and others. I doubt if it were not best to sell them all for slaves. For in truth what I have now done is as if a man were to kill the father and then spare the children. Thee, who wert something more than a father to thy people, I have seized and carried off, and to that people I have entrusted their city. Can I then feel surprise at their rebellion ?' Thus did Cyrus open to Croesus his thoughts; whereat the latter, full of alarm lest Cyrus should lay Sardis in ruins, replied as follows:-'0 King, thy words are reasonable, but do not, I beseech thee, give full vent to thine anger nor destroy an ancient city, guiltless alike of the past and of the present trouble. I caused the former, and in my own person now pay the penalty.'
III. TRINITY TERM, 1902. He soon had the pleasure of fighting the king of the island of Cyprus, because he allowed his subjects to pillage some of the English troops who were shipwrecked on the shore; and easily conquering this poor monarch, he seized his only daughter, and put the king himself into silver fetters. He then sailed away again with his mother, sister, wife and the captive princess, and soon arrived before a certain town, which the French king was besieging with his fleet. This king's army, however, had been wasted by the plague and thinned by the swords of the enemy, whose numerous army was at that time gallantly defending the place. The English and French kings were jealous of each other, and discord also reigned between the disorderly and violent soldiers of the two nations. Hence they could not at first agree to make the assault ; but when they did, the enemy promised to yield the town.
IV. SEPTEMBER, 1902. Then these two men went and showed themselves to the king, and told him how it had come to pass that they were thus treated. Darius feared lest it was by common consent that the deed had been done; he therefore sent for them all in turn, because he wished to ascertain whether they approved what Intaphernes had done. When he had heard their answers, he laid hands on Intaphernes, his children and all his kindred ; suspecting that he and his friends were about to revolt. When all had been seized and put in chains, the wife of Intaphernes came and stood at the palace gates, weeping and wailing sore. So Darius after a while pitied her, and bade a messenger go to her and say, 'Lady, King