Plane trigonometry, by S.L. Loney.

[Exs. XXXVII.] PROPERTIES OF TRIANGLES.

247

29. If a circle be drawn touching the inscribed and circumscribed
circles of a triangle and the side BC externally, prove that its radius is
A A
tan2 A
a 2
30. If a, b, c be the radii of three circles which touch one another
externally and rl and r2 be the radii of the two circles that can be drawn
to touch these three, prove that
1 1 2 2 2
+ — -=-+-+-
2'1?'2 a b c
31. If A0 be the area of the triangle formed by joining the points of
contact of the inscribed circle with the sides of the given triangle, whose
area is A, and A1, A2, and A3 the corresponding areas for the escribed
circles, prove that
A+ A2+ A3- A0= 2A.
32. If the bisectors of the angles of a triangle ABC meet the opposite
sides in A', B', and C', prove that the ratio of the areas of the triangles
A'B'C' and ABC is
A. B. C A-B B-C C-A
2 sin - sin sin -: cos 2 cos 2 cos --
2 2 2 2 2 2
33. Through the angular points of a triangle are drawn straight
lines which make the same angle a with the opposite sides of the triangle;
prove that the area of the triangle formed by them is to the area of the
original triangle as 4 cos2a: 1.
34. Two circles, of radii a and b, cut each other at an angle 0.
Prove that the length of the common chord is
2ab sin 0
/a2 + b2 + 2ab cos 0
35. Three equal circles touch one another; find the radius of the
circle which touches all three.
36. Three circles whose radii are a, b and c touch one another and
the tangents at their points of contact meet in a point; prove that the
distance of this point from either of their points of contact is
/ abc '
a+ b+c]