QUESTIONS FOR REVISION ON BOOK IV.

1. With what difference of meaning is the word inscribed used in the following cases?

(i) a triangle inscribed in a circle ;

(ii) a circle inscribed in a triangle.

2. What is meant by a cyclic figure? Shew that all triangles are cyclic.

What is the condition that a quadrilateral may be cyclic?

Shew that cyclic parallelograms must be rectangular.

3. Shew that the only regular figures which may be fitted together so as to form a plane surface are (i) equilateral triangles, (ii) squares, (iii) regular hexagons.

2(n-2)

4. Employ the first Corollary of I. 32 to shew that in any regular polygon of n sides each interior angle contains right angles ?

5.

The bisectors of the angles of a regular polygon are concurrent. State the method of proof employed in this and similar theorems. 6. Shew that

(i) all squares inscribed in a given circle are equal; and
(ii) all equilateral triangles circumscribed about a given
circle are equal.

7. How many circles can be described to touch each of three given straight lines of unlimited length?

8.

(i) when no two of the lines are parallel ;

(ii) when two only are parallel ;

(iii) when all three are parallel.

Prove that the greatest triangle which can be inscribed in a circle on a diameter as base, is one-fourth of the circumscribed square.

9. The radius of a given circle is 10 inches: find the length of a side of

(i) the circumscribed square ;

(ii) the inscribed square;

(iii) the inscribed equilateral triangle;
(iv) the circumscribed equilateral triangle;
(v) the inscribed regular hexagon.

Shew also that the areas of these figures are 200, 75√3, 300√3, and 150√√3 square inches.

[20 inches.] √2 inches.]

[10√3 inches.] [20√√3 inches.]

[10 inches.]

respectively 400,

THEOREMS AND EXAMPLES ON BOOK IV.

I. ON THE TRIANGLE AND ITS CIRCLES.

1

1. D, E, F are the points of contact of the inscribed circle of the triangle ABC, and D1, E, F, the points of contact of the escribed circle, which touches BC and the other sides produced: a, b, c denote the length of the sides BC, CA, AB; s the semi-perimeter of the triangle, and r, r, the radii of the inscribed and escribed circles.

2. In the triangle ABC, I is the centre of the inscribed circle, and 1,2,3 the centres of the escribed circles touching respectively the sides BC, CÅ, AB and the other sides produced.

D3

B

D2

Prove the following properties:

(i) The points A, I, I1 are collinear: so are B, 1, 12; and C, 1, 13. The points 12, A, 13 are collinear; so are 13, B, I1; and

(ii)

11, C, 12.

(iii) another.

The triangles BIC, CIA, AI,B are equiangular to one

(iv) The triangle 23 is equiangular to the triangle formed by joining the points of contact of the inscribed circle.

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

(vi) The four circles, each of which passes through three of the points 1, 11, 12, 13, are all equal.

3. With the notation of page 297, shew that in a triangle ABC, if the angle at C is a right angle,

r=s-c; r1=s-b; r2=s-a; rz=s.

4. With the figure given on page 298, shew that if the circles whose centres are 1, 11, 12, 13 touch BC at D, D1, D2, D3, then

(i) DD2=D1Dg=b. (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. 243.]

6. Given the base and vertical angle of a triangle, find the locus of the centre of the inscribed circle. [See Ex. 36, p. 246.]

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 vertical 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 I1: 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. is the centre of the circle inscribed in a triangle, and 11, 12, 13 the centres of the escribed circles; shew that I, II, II, 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, I 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 magni-
tude 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 in-
scribed 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. 276.]

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, r1 are the radii of the inscribed circle and of the
escribed circle which touches BC: shew that r.r1 =BD. 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.