DEF. V. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment as the greater segment is to the less.

PROPOSITION XXX. PROBLEM. (Eucl. VI. 30.)

To cut a straight line in extreme and mean ratio.

Let AB be the given st. line.

It is required to cut AB in extreme and mean ratio.
Divide AB in the pt. C, so that rect. AB, BC

=

sq. on AC. II. 11.

.. AB is to AC as AC is to BC,

VI. 17.

and .. AB is cut in extreme and mean ratio in C. Def. 5.

Q. E. F.

Ex. 1. If two diagonals of a regular pentagon be drawn to cut one another, they cut one another in extreme and mean ratio.

Ex. 2. If the radius of a circle be cut in extreme and mean ratio, the greater segment will be equal to the side of a regular decagon described in the circle.

PROPOSITION XXXI. THEOREM. (Eucl. VI. 32.)

If two triangles, SIMILARLY SITUATED, which have two sides of the one proportional to two sides of the other, be joined at one angle, so as to have their homologous sides parallel, each to each, the remaining sides must be in a straight line.

B

D

Let the As ABC, DCE be similarly situated, having the sides BA, AC proportional to CD, DE, and let BA be to CD, and AC to DE;

Then BA is to AC as CD is to DE, and ▲ BAC = 2 CDE,

L

..▲ ABC is equiangular to ▲ DCE.

.. 4 ACB = ▲ DEC;

VI. 6.

VI. Def. 1.

and.. 48 ACB, ACE together =LS ACE, DEC together,

= two right angles.

I. 29.

.. BC and CE are in the same st. line.

I. 14.

1. Two common tangents to two circles meet at A. If the diameter of the smaller circle, the distance between the centres, and the diameter of the larger circle, be in the ratio of 1, 2, 3, prove that the distance from A to the centre of each circle is equal to the diameter of that circle.

2. Straight lines are drawn through the angular points of a triangle, parallel to the opposite sides, and through the angular points of the triangle thus formed straight lines are drawn, parallel to its opposite sides, and so on; show that all these triangles are similar to the original triangle, and that any one of them has its sides bisected by the angular points of the preceding triangle.

3. If a point be taken within an equilateral triangle, the perpendiculars drawn from it to the three sides are together equal to the perpendicular drawn from one of the angles to the opposite side.

4. Upon AB as base two triangles ABC, ABD are described, and a line cutting CA is drawn parallel to CD. From the points where this line meets AC, AD, lines are drawn to meet CB, DB, and parallel to the base. Shew that these lines are equal.

5. If O be the centre, and AB the diameter of a circle, and if on the radius 40 a circle be described, then the circumference of this circle will bisect any chord, drawn through it from A to meet the exterior circle.

6. On a given base describe a triangle, having a given vertical angle, and one of its sides double of the other.

7. From a point E in the common base of two triangles ACB, ADB, straight lines are drawn parallel to AC, AD, meeting BC, BD in F and G. Shew that the lines joining F, G and C, D will be parallel.

8. From the angular points, of a triangle ABC, straight lines AD, BE, CF, are drawn perpendicular to the opposite sides

and terminated by the circumscribing circle; if L be the point of their intersection, shew that LD, LE, LF are bisected by the sides of the triangle.

9. If D and E be points in the sides of a triangle ABC, such that AD and AE are respectively the third parts of AB and AC, shew that BE and CD cut one another in a point of quadrisection.

10. In AB, AC, two sides of a triangle, are taken points D, E; AB, AC are produced to F, G, such that BF=AD, and CG=AE: and BG, CF, FG are joined, the two former meeting in H. Show that the triangle FHG is equal to the triangles BHC, ADE together.

11. If the angle, between the internal bisector of the angle of a triangle and the base, be equal to the angle between the external bisector and the greater side produced, a perpendicular on this side through the vertex will bisect the segment of the base between the internal and external bisectors.

12. Triangles on equal bases and between the same parallels will have equal areas cut off by a line parallel to their bases.

13. From A, B, the extremities of the diameter of a circle, lines ACE, BCD, are drawn through a point C, on the circumference, to points E and D, such that EB and DA touch the circle. Shew that ED is parallel to the tangent at C.

14. Draw a straight line cutting two concentric circles, so that the part of it which is intercepted by the circumference of the greater may be four times as great as the part intercepted by the circumference of the less.

15. Shew how to inscribe a rectangle DEFG in a triangle ABC, so that the angles D, E may be in AB, AC respectively, the side FG coincident with the base, and the area of the rectangle be equal to half that of the triangle.

16. If the bisectors of the opposite angles A, C, of a quadrilateral figure ABCD, intersect on the diagonal BD, then will the bisectors of the angles B, D meet on AC.

17. Two sides of a quadrilateral described about a circle are

parallel; if the points of contact divide the other two sides proportionally, they are equally inclined to the first two.

18. If two triangles, on the same base, have their vertices joined by a straight line, which meets the base, or the base produced, shew that the parts of this line, between the vertices of the triangles and the base, are in the same ratio to each other as the areas of the triangles.

19. From any point P, in the circumference of a circle, whose centre is O, perpendiculars PM, PN, are let fall on two radii OA, OB, and are produced both ways to meet the circumference of the circle in C, D, and the straight lines OA, OB, in E, F respectively. Shew that the three straight lines CD, MN, EF, are parallel to one another.

20. If the angles B, C, of the triangle ABC, be respectively equal to the angles D, E, of the triangle ADE, and the angles B, E, of the triangle ABE, to the angles D, C, of the triangle ADC, then these pairs of triangles shall be respectively equal to each other; and if BE, CD, intersect in F, the triangles BFD, CFE, shall also be similar.

21. If, from the extremities of the diameter of a semicircle, perpendiculars be let fall on any line cutting the semicircle, the parts intercepted between those perpendiculars and the circumference are equal.

22. In a given circle place a chord, parallel to a given chord, and having a given ratio to it.

23. ABC is an equilateral triangle. Through C a line is drawn at right angles to AC, meeting AB produced in D, and a line through A parallel to BC in E. Through K, the middle point of AB, lines are drawn respectively parallel to AE, AC, and meeting DE in F and G. Prove that the sum of the squares on KG and FG is equal to three times the square on FE.

24. Find a point in the base of a right-angled triangle produced such that the line drawn from it to the angular point opposite to the base, shall be to the base produced as the perpendicular to the base itself.