Track / Stage 5 / 248 of 400 #848 of 2000

Problem 848

AIME late

Geometry Difficulty 5.6 Prove it

8. In $\triangle A B C$ , prove:
(1) $s=b \cos ^{2} \frac{C}{2}+c \cos ^{2} \frac{B}{2}$ ;
(2) $s=a+b \sin ^{2} \frac{C}{2}+c \sin ^{2} \frac{B}{2}$ ;
(3) $s=4 R \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}$ ;
(4) $s=r\left(\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}\right)$ .

This one wants a proof. Work it on paper, read the official solution, then mark yourself honestly — the ladder only means something if the record is true.

Official solution

8. (1) Reduce the order and use the projection theorem.
(2) Use $\sin ^{2} x+\cos ^{2} x=1$ to transform into (1).
(3) $s=\frac{\Delta}{r}$ , proved by the conclusion of the 11th sub-question of the 1st question.
(4) $s=\frac{\Delta}{r}$ , from the conclusion of the 2nd sub-question of the 1st question, we know $\cot \frac{A}{2} \cot \frac{B}{2} \cot \frac{C}{2}=\cot \frac{A}{2}+\cot \frac{B}{2}+\cot \frac{C}{2}$ .

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.