MHB Please find the side length of an equilateral triangle

[Albert1]

P is an inner point of equilateral triangle ABC ,
given PC=3, PA=4,PB=5 ,please find the side length of the equilateral triangle

 

[caffeinemachine]

P is an inner point of equilateral triangle ABC ,
given PC=3, PA=4,PB=5 ,please find the side length of the equilateral triangle

Rotate the trianlge by 60 degrees about the point C. Let $P'$ be the new position of $P$. Then $\angle P'PA=90$. Thus we get a triangle $CPA$ with $\angle CPA=150$ and $PC=3, PA=4$. To find $AC$.

 

[Albert1]

solution :

View attachment 1229

 

[caffeinemachine]

solution :

View attachment 1229

Brilliant solution!