20. Take any three points, A, B, C in the circumference of a circle. Join AB, BC, AC, and draw AD. AE parallel to the tangents at B and C, and meeting BC produced if necessary in D and E; and prove that the segments BD and EC are to each other in the duplicate ratio of AB to AC.

21. If the diameter AB of a circle be divided into an odd number

(n) of equal parts, and C and D be the 1 (2-1) th

th

and (n+1)" divisions;

2

and AEC, AFD, CGB, DHB be semicircles: shew that the perimeter of the figure AECGBHDF is equal to that of the circle, and its area an nth part of the area of the circle.

22. The circle inscribed in a square is equal to four equal circles touching one another and the sides of that square internally.

23. If AB be a circular arc, centre O, and AD be drawn perpendicular to BO, and the arc AC taken equal to AD, then the sector BOC equals the segment ACB.

24. Let AB and DC be two diameters of a given circle, at right angles to each other; AEB a circular arc described with radius DB or DA; prove that the area of the lune AEBC = area of triangle ADB.

25. If two points B, D, be taken at equal distances from the ends of the arc of a quadrant, and perpendiculars BG, DH be drawn to the extreme radius; the space BGHD shall be equal to the sector BOD.

26. ABC is an isosceles right-angled triangle. On BC is described a semicircle BDEC, and BFC is a circle whose radius is AB and centre A. The segment BCF is equal to the segments BAD, ACE.

27. If a semicircle be described on the hypothenuse AB of a right-angled triangle ABC, and from the centre E, the radius ED be drawn at right angles to AB, shew that the difference of the segments on the two sides equals twice the sector CED.

28. If semicircles be described upon the sides of a right-angled triangle on the interior, the difference between the sum of the circular segments thus standing upon the exterior of the sides and segments of the base equals the space intercepted by the circumferences described on the sides.

29. Semicircles are described upon the radii CA, CB of a quadrant, and intersect each other in a point D, shew (a) That the points B, D, A are in one straight line. (b) That the area common to both semicircles is equal to the area without them. (c) That the remaining areas of the two semicircles are equal, each is one fourth of the square on AC. 30. If two parallel planes cut a sphere so that the sections are equal, they are equidistant from the centre.

31. Shew that all lines drawn from an external point to touch a sphere are equal to one another; and thence prove that if a tetrahedron can have a sphere inscribed in it touching its six edges, the sum of every two opposite edges is the same.

32. If two equal circles cut one another in the diameter, and a plane cut them perpendicularly to the same diameter, the points of section of this plane with the circumferences, are in a circle.

33. In any polyhedron having different faces, some with an even, some with an odd number of sides: shew that the number of those faces which have an odd number of sides is necessarily an even number.

34. The angles of inclination of the faces of a regular tetrahedron and of a regular octahedron are supplementary to each other.

1 Sid.,30.,43. Qu.,34.
Trin.,40.

2 Emm.,22.,35. Sid.,30.
Trin. 37.

3 S. H.,17.
Trin. ,24.
,37. Qu.,25. Emm.,27.
Cath. 29. Pem. 39.
Sid. ,40. Chr.,45.
4 Emm.,21. Qu.,23.
,40.,42. Trin. ,26.,27.
,29. C. C.,30.
Pem.,32.,38.

8 Ki.,45. Cath.,30.
9 Trin.,32.,37.
10 Qu.,31. Cath. ,35.
Emm.,35. Sid.,38.
B. S.,40. Trin. ,27.

11 Trin.,34.
12 Trin. ,44.

13 Qu.,32. Jes.,36.
14 Chr. 42.
15 Qu.,19.
16 Trin.,38.
17 Trin.,36.
18 Trin. ,43.