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.

Guest Mar 16, 2020

#1**+1 **

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 **

Dragan Mar 16, 2020

#2**0 **

Dragan, could you please give your explanation and format your answer as a root? Please and thank you!

Guest Mar 25, 2020

#3**0 **

\(4\sqrt7+2\text{ is the answer}\)(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)

Guest Mar 25, 2020

#4**+1 **

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

CPhill Mar 25, 2020