$[ABP]+[BCP] = \frac{8*s*\sin(60-x)+8s*\sin(x)}{2} \leq^{Jensen} \frac{16s*\sin(30)}{2} =4s$ achieved when $x=30$ $15^2 =8^2+s^2-2*8*s*\cos(30)$ $s=4\sqrt{3}+\sqrt{209}$ $[ABP]+[BCP] = \frac{8*s*\sin(60-x)+8s*\sin(x)}{2} \leq^{Jensen} \frac{16s*\sin(30)}{2} =4s=4(4\sqrt{3}+\sqrt{209})$ is there something wrong ?