MHB Find the Max of $PC$ in $\triangle ABC$

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$\triangle ABC$ with $AB=BC=CA$
if another point $P$ and $PA=2, \,\, PB=3$
please find :$max(PC)$
 
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Albert said:
$\triangle ABC$ with $AB=BC=CA$ if another point $P$ and $PA=2, \,\, PB=3$ please find :$max(PC)$

my solution:
 

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Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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