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[Solved] Triangle and logarithm together

 
Triangles & Diagonals
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June 15, 2021 3:28 pm
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Staff 
(@armor)
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Let A, B, C be the three interior angles of  \Delta ABC.  lgA, lgB, lgC form an arithmetic sequence. Find the range of B.

 

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June 15, 2021 3:28 pm
Topic starter
Staff 
(@armor)
Admin
  Honorable Member
317 Posts
156 153 0

From the given condition, we have \small lgA+lgC=2lgB,  then  \small B^2=AC.  Thus \small C>B>A  and    \small B<\frac{\Pi }{2}.

Hence    \small [\Pi -(A+C)]^2=AC\leq (\frac{A+C}{2})^2.   Since 

\small \frac{A+C}{2}\leq B<\frac{\Pi }{2}\Rightarrow \Pi -(A+C)\geq \frac{A+C}{2}\Rightarrow A+C\leq \frac{2\Pi }{3}\Rightarrow B\geq \Pi -\frac{2\Pi }{3}=\frac{\Pi }{3}.

Therefore  \small \frac{\Pi }{3}\leq B<\frac{\Pi }{2}.

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