MHB Please find the side length of an equilateral triangle

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To find the side length of an equilateral triangle ABC with an inner point P, given distances PA=4, PB=5, and PC=3, a rotation of the triangle by 60 degrees around point C is utilized. This rotation positions point P' such that angle P'PA becomes 90 degrees, creating triangle CPA with angle CPA measuring 150 degrees. Using the known lengths and angles, the side length AC can be calculated. The solution effectively demonstrates how geometric transformations can simplify complex problems. The method highlights the relationship between the inner point and the triangle's vertices.
Albert1
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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
 
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Albert said:
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 $AC$.
 

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