S-Area-of-AB-C-in-AB-C-a-2-4S-1-2-cot-B-cot-C-a-2-4S-a-2-4-1-2-bc-sin-A-4R-2-sin-2-A-8R-2-sin-B-sin-

Question Number 201134 by mnjuly1970 last updated on 30/Nov/23

S : A r e a o f A \overset{Δ}{B C}

i n A \overset{Δ}{B C} : \frac{a^{2}}{4 S} \overset{?}{=} \frac{1}{2} (c o t (B) + c o t (C))

\frac{a^{2}}{4 S} = \frac{a^{2}}{4 (\frac{1}{2} b c s i n (A))} = \frac{4 R^{2} {s i n}^{2} (A)}{8 R^{2} s i n (B) s i n (C) s i n (A)}

= \frac{s i n (A)}{2 s i n (B) s i n (C)}

\overset{A + B + C = π}{=} \frac{s i n (B) c o s (C) + c o s B s i n (C)}{2 s i n (B) s i n (C)}

= \frac{1}{2} (c o t (B) + c o t (C))