Half-Angle Tangents and the Circumcenter

★ 7/10plane geometry trigonometry Arithmetic Progression Triangle Geometry vectors

Problem

In $\triangle ABC$ ,

\tan\frac{A}{2}\tan\frac{B}{2} = \frac13,

with the usual side notation.

1. Prove that $a, b, c$ can be rearranged into an arithmetic progression. 2. Let $O$ be the circumcenter, the perimeter be $12$ , and $c - a = 1$ . Find

m = \overrightarrow{OA}\cdot\overrightarrow{OB} + \overrightarrow{OB}\cdot\overrightarrow{OC} + \overrightarrow{OC}\cdot\overrightarrow{OA}.

Answer

Solution

Difficulty	7/10
Topics	plane geometry, trigonometry, Arithmetic Progression, Triangle Geometry, vectors

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