Lengths of sides of ABC form an arithmetic progression

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Lengths of sides of ABC form an arithmetic progression

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Source: Romanian MO 2010 Grade 9

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#1 h Aug 6, 2012, 1:17 PM • 2 Y

Y by Adventure10, Mango247

Prove that there is a similarity between a triangle $ABC$ and the triangle having as sides the medians of the triangle $ABC$ if and only if the squares of the lengths of the sides of triangle $ABC$ form an arithmetic sequence.

Marian Teler & Marin Ionescu

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#2 h Aug 6, 2012, 3:45 PM • 2 Y

Y by Adventure10 and 1 other user

If part : Wlog $a^2+b^2=2c^2$
Then it's easy to see $\frac {m_a}{b}=\frac {m_b}{a}=\frac {m_c}{c}=\frac {\sqrt {3}}{2}$
So if part is done.
Suppose only if part is not true.

Now we've $k^2(a^2+b^2+c^2)=\frac {3}{4}(a^2+b^2+c^2)$
So finally we get $2(b^2+c^2)-a^2=3b^2,3c^2,3a^2$
So in all cases it forms a a.p .... hence only if part is also true, done.

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