Plane trigonometry, by S.L. Loney.

248 TRIGONOMETRY. [Exs. XXXVII.]
37. In the sides BC, CA, AB are taken three points A', B', C' such that
BA': A'C=CB': B'A=AC': C'B=m: n;
prove that if AA', BB', and CC' be joined they will form by their intersections a triangle whose area is to that of the triangle ABC as
(m - n)2: m2 +mn + n2.
38. The circle inscribed in the triangle ABC touches the sides BC,
CA, and AB in the points A1, B1, and C1 respectively; similarly the
circle inscribed in the triangle AlBlC1 touches the sides in A2, B2, C2
respectively and so on; if AnBnC, be the nth triangle so formed, prove
that its angles are
-+(-2)-(A 3), +(-2)-(B ),
Hand pvt ter +(-2)-n(c- ).
Hence prove that the triangle so formed is ultimately equilateral.
39. ABCI is the triangle formed by joining the feet of the perpendiculars drawn from ABC upon the opposite sides; in like manner
A2B2C2 is the triangle obtained by joining the feet of the perpendiculars
from Al, B1, and C1 on the opposite sides and so on. Find the values of
the angles A,, B,, and C, in the nth of these triangles.