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# Geometry problem.

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Let be P the centroid of triangle ABC. If triangle ABP is an equilateral triangle with a side length of 2, then find the perimeter of triangle ABC.

Mar 16, 2020

#1
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Let be P the centroid of triangle ABC. If triangle ABP is an equilateral triangle with a side length of 2, then find the perimeter of triangle ABC.

AB = 2

PF = sqrt(3)

CF = PF * 3

AC = BC = sqrt[(AF)² + (CF)²]

Perimeter of triangle ABC = AB + AC + BC  Mar 16, 2020
edited by Dragan  Mar 16, 2020
edited by Dragan  Mar 25, 2020
edited by Dragan  Mar 25, 2020
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Mar 25, 2020
#3
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$$4\sqrt7+2\text{ is the answer}$$(I assume this is an AoPS challenge problem and thus you don't need any information. If you do, then remember how 1:2 ratios work with centroids)

Mar 25, 2020
#4
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Using Dragan's  illustration.....

AP    =  2

And

sin (60°)  = PF /  AP

sin(60°)  = PF / 2  ⇒   PF  = √3

But CF is a median of  triangle  ABC  and  the PF  is 1/3 of the altitude FC  = 3√3   = √27

And  using the Pythagorean Theorem

CB  =√ [ FC^2  + FB^2 ]=   √ [27 + 1 ]  =   √28  =   2√7  =   CA

So  the perimeter  =  CB + CA  + AB =    2√7  + 2√7 +  2  =    4√7  + 2   Mar 25, 2020