EXERCISES ON PROPOSITIONS 11-16.

1. Express in terms of a right angle the magnitude of an angle of the following regular polygons:

(i) a pentagon, (ii) a hexagon, (iii) an octagon,

(iv) a decagon, (v) a quindecagon.

2. The angle of a regular pentagon is trisected by the straight lines which join it to the opposite vertices.

3. In a polygon of n sides the straight lines which join any angular point to the vertices not adjacent to it, divide the angle into n 2 equal parts.

4. Shew how to construct on a given straight line

(i) a regular pentagon, (ii) a regular hexagon, (iii) a regular octagon.

5. An equilateral triangle and a regular hexagon are inscribed in a given circle; shew that

(i) the area of the triangle is half that of the hexagon;

(ii) the square on the side of the triangle is three times the square on the side of the hexagon.

6. ABCDE is a regular pentagon, and AC, BE intersect at H: shew that

(i) AB=CH=EH.

(ii) AB is a tangent to the circle circumscribed about the triangle BHC.

(iii) AC and BE cut one another in medial section.

7. The straight lines which join alternate vertices of a regular pentagon intersect so as to form another regular pentagon.

8. The straight lines which join alternate vertices of a regular polygon of n sides, intersect so as to form another regular polygon of n sides.

If n=6, shew that the area of the resulting hexagon is one-third of the given hexagon.

9. By means of iv. 16, inscribe in a circle a triangle whose angles are as the numbers 2, 5, 8.

10. Shew that the area of a regular hexagon inscribed in a circle is three-fourths of that of the corresponding circumscribed hexagon.

THEOREMS AND EXAMPLES ON BOOK IV,

I. ON THE TRIANGLE AND ITS CIRCLES,

1. D, E, F are the points of contact of the inscribed circle of the triangle ABC, and D, E, F, the points of contact of the escribed circle, which touches BC and the other sides produced: a, b, c denote the lengths of the sides BC, CA, AB; s the semi-perimeter of the triangle, and r, r1 the radii of the inscribed and escribed circles.

2. In the triangle ABC, I is the centre of the inscribed circle, and 12, the centres of the escribed circles touching respectively the sides BC, CA, AB and the other sides produced.

D3

B

D2

Prove the following properties :

and C, I, 13.

(i) The points A, I, I, are collinear; so are B, I, (ii) The points 12, A, 13 are collinear; so are Ig, B, I1; and I, C, I.

another.

(iii) The triangles BIC, CIA, AIB are equiangular to one

(iv) The triangle 1, is equiangular to the triangle formed by joining the points of contact of the inscribed circle.

(v) of the four points I, 11, 12, I, each is the orthocentre of the triangle whose vertices are the other three.

points!

(vi) The four circles, each of which passes through three of the 1, are all equal.

3. With the notation of page 277, shew that in a triangle ABC, if the angle at C is a right angle,

r=8-c; r1s-b; r2=s-a; r3=s.

4. With the figure given on page 278, shew that if the circles whose centres are I, I, I, I, touch BC at D, D1, D2, D3, then

DD2=D1Dg=b.

(i)
(iii) D2D=b+c.

(ii) DD-D1D2 = c.
(iv) DD1=b~c.

5. Shew that the orthocentre and vertices of a triangle are the centres of the inscribed and escribed circles of the pedal triangle. [See Ex. 20, p. 225.]

6. Given the base and vertical angle of a triangle, find the locus of the centre of the inscribed circle. [See Ex. 36, p. 228.]

7. Given the base and vertical angle of a triangle, find the locus of the centre of the escribed circle which touches the base.

8. Given the base and vertical angle of a triangle, shew that the centre of the circumscribed circle is fixed.

9. Given the base BC, and the vertical angle A of a triangle, find the locus of the centre of the escribed circle which touches AC.

10. Given the base, the vertical angle, and the radius of the inscribed circle; construct the triangle.

11. Given the base, the vertical angle, and the radius of the escribed circle, (i) which touches the base, (ii) which touches one of the sides containing the given angle; construct the triangle.

12. Given the base, the vertical angle, and the point of contact with the base of the inscribed circle; construct the triangle.

13. Given the base, the vertical angle, and the point of contact with the base, or base produced, of an escribed circle; construct the triangle.

14. From an external point A two tangents AB, AC are drawn to a given circle; and the angle BAC is bisected by a straight line which meets the circumference in I and 1: shew that I is the centre of the circle inscribed in the triangle ABC, and I, the centre of one of the escribed circles.

15. I is the centre of the circle inscribed in a triangle, and 11, 12, la the centres of the escribed circles; shew that II1, 112, 113 are bisected by the circumference of the circumscribed circle.

16. ABC is a triangle, and 12, 13 the centres of the escribed circles which touch AC, and AB respectively: shew that the points B, C, 12, 13 lie upon a circle whose centre is on the circumference of the circle circumscribed about ABC,

17. With three given points as centres describe three circles touching one another two by two. How many solutions will there be?

18. Two tangents AB, AC are drawn to a given circle from an external point A; and in AB, AC two points D and E are taken so that DE is equal to the sum of DB and EC: shew that DE touches the circle.

19. Given the perimeter of a triangle, and one angle in magnitude and position: shew that the opposite side always touches a fixed circle. 20. Given the centres of the three escribed circles; construct the triangle.

21. Given the centre of the inscribed circle, and the centres of two escribed circles; construct the triangle.

22. Given the vertical angle, perimeter, and the length of the bisector of the vertical angle; construct the triangle.

23. Given the vertical angle, perimeter, and altitude; construct the triangle.

24. Given the vertical angle, perimeter, and radius of the inscribed circle; construct the triangle.

25. Given the vertical angle, the radius of the inscribed circle, and the length of the perpendicular from the vertex to the base; construct the triangle.

26. Given the base, the difference of the sides containing the vertical angle, and the radius of the inscribed circle; construct the triangle. [See Ex. 10, p. 258.]

27. Given a vertex, the centre of the circumscribed circle, and the centre of the inscribed circle, construct the triangle.

28. In a triangle ABC, I is the centre of the inscribed circle; shew that the centres of the circles circumscribed about the triangles BIC, CIA, AIB lie on the circumference of the circle circumscribed about the given triangle.

29. In a triangle ABC, the inscribed circle touches the base BC at D; and r, r, are the radii of the inscribed circle and of the escribed circle which touches BC: shew that r. rBD. DC.

30. ABC is a triangle, D, E, F the points of contact of its inscribed circle and D'E'F' is the pedal triangle of the triangle DEF: shew that the sides of the triangle D'E'F' are parallel to those of ABC.

31. In a triangle ABC the inscribed circle touches BC at D. Shew that the circles inscribed in the triangles ABD, ACD touch one another.