# 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 $$a^{2}+b^{2}+c^{2}+d^{2}$$.

×