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In triangle ABC, the quantities tan(A/2), tan(B/2), and tan(C/2) are in Harmonic Progression. Which of the following is/are correct? (Symbols have their usual meanings.)

1. a, b, c are in Arithmetic Progression
2. sin(A), sin(B), sin(C) are in Harmonic Progression
3. tan(A/2) * tan(C/2) = 1/3
4. a, b, c are in Geometric Progression

Correct answer: a, b, c are in Arithmetic Progression

Solution

Since tan(A/2) = r/(s-a), the terms being in HP means (s-a), (s-b), (s-c) are in AP, which directly implies a, b, c are in AP. Also from the AM property of the triangle, tan(A/2)*tan(C/2) = r²/((s-a)(s-c)); for a, b, c in AP one can show this equals 1/3.

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