These points will make you crazy!

Geometry Level 5

Let \(A_{1}\), \(A_{2}\), \(A_{3}\), \(A_{4}\) and \(A_{5}\) be five points in a plane whose coordinates are \((1,-1)\), \((2,-3)\), \((3,-2)\), \((-10,-4)\) and \((4,10)\) respectively.

\(A_{1}A_{2}\) is bisected at \(B_{1}\); \(\ B_{1}A_{3}\) is divided at \(B_{2}\) in the ratio \(1:2\); \( \ B_{2}A_{4}\) is divided at \(B_{3}\) in the ratio \(1:3\) and \(B_{3}A_{5}\) is divided at \(B_{4}\) in the ratio \(1:4\).

Given that \(B_{4}\) is the circumcenter of \(\triangle KLM\) with coordinates of \(K=(\sin \theta,\cos \theta)\), \(L=(a,b)\) and \(M=(c,d)\).

Find the value of \(a^{2}+b^{2}+c^{2}+d^{2}\).

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