Plane trigonometry, by S.L. Loney.

242 TRIGONOMETRY.
If OF be perpendicular to AB, we have
zOAF= 90- z AOF= 90 -C.
Also zPAL = 90 - C.
z. OAP=A-z OAF- PAL
=A - 2 (90~ - C)= A + 2- 180c
=A + 2C- (A +B +C)=C- B.
Also OA = R, and, by Art. 209,
PA = 2Rcos A... OP2= OA2 + PA2- 20A. PA cos OAP
= R2 + 4R2 cos2 A - 4R2 cos A cos (C- B)
= R2 + 4R2 cos A [cos A - cos (C- B)]
= R2 - 4R2 cos A [cos (B + C) + cos ( - B)]
(Art. 72),
= R2 - 8R2 cos A cos B cos C..'. OP = R V1 -8 cos A cos B cos C.

*217. To find the distance between the circumc
and the incentre.
Let 0 be the circumcentre and
OF perpendicular to AB.
Let I be the incentre and IE F
perpendicular to AC.
Then, as in the last article, B
z OAF= 900 -C... OAI =zIAF-z OAF
A (9o - C) A A+B+C C-B
2 90- - 2^ = 2-

entre