Plane trigonometry, by S.L. Loney.

246

TRIGONOMETRY.

[Exs. XXXVII.]

20. DEF is the triangle formed by joining the points of contact of
the incircle with the sides of the triangle ABC; prove that
A B C
(1) its sides are 2r cos, 2r cos, and 2r cos -
2 2' 2'
7r A d r B v r C
(2) itsanglesare 2 2 -2 and 2 C
~- - -2'2 - 2
2S3 1 r
and 1(3) its area is -S, i.e. - S.
21. D, E, and F are the middle points of the sides of the triangle
ABC; prove that the centroid of the triangle DEF is the same as that of
ABC and that its orthocentre is the circumcentre of ABC.
In any triangle ABC, prove that
22. The perpendicular from A divides BC into portions which are
proportional to the cotangent of the adjacent angles, and that it divides
the angle A into portions whose cosines are inversely proportional to the
adjacent sides.
23. The median through A divides it into angles whose cotangents
are 2 cot A + cot C and 2 cot A + cot B, and makes with the base an angle
whose cotangent is 2 (cot C - cot B).
24. The distance between the middle point of BC and the foot of the
b2- C2
perpendicular from A is 2
2a
25. O is the orthocentre of a triangle ABC; prove that the radii of
the circles circumscribing the triangles BOC, COA, AOB and ABC are
all equal.
26. AD, BE and CF' are the perpendiculars from the angular points
of a triangle ABC upon the opposite sides; prove that the diameters of
the circumcircles of the triangles AEF, BDF and CDE are respectively
a cot A, b cot B, and c cot C, and that the perimeters of the triangles DEF
and ABC are in the ratio r: R.
27. Prove that the product of the distances of the incentre from the
angular points of a triangle is 4Rr2.
28. The triangle DEF circumscribes the three escribed circles of the
triangle ABC; prove that
EF FD DE
acos A b cos B- c cos C'