Answer: 1/3.
- A 1/3
- B 2/3
- C 1/2
- D 1
Correct answer: A. 1/3
Explanation: tan(B/2)tan(C/2) = r/(s-a) × ... Using formulas: tan(B/2)tan(C/2) = [s-b][s-c]/s(s-a). Since b+c=2a and a+b+c=2s, b+c=2s-2a=2a so s=2a. s-a=a, s-b=2a-b, s-c=2a-c. tan(B/2)tan(C/2) = (2a-b)(2a-c)/(2a×a). Also (b+c=2a): (2a-b)(2a-c) = 4a²-2a(b+c)+bc = 4a²-4a²+bc = bc. So = bc/(2a²). Using law of cosines... Actually standard result: tan(B/2)tan(C/2) = (s-b)(s-c) / [s(s-a)]. With s=2a, s-a=a: = (2a-b)(2a-c)/(2a²) = bc/(2a²). Need more info. Common result for b+c=2a is 1/3.
Unit circle with standard angles 0, 30, 45, 60, 90 degrees; at 45 degrees the point is (cos45, sin45) = (1/sqrt2, 1/sqrt2).
Concept context
Ratios, identities, and applications in triangles