Questions on the use of algebraic techniques to prove geometric theorems.

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Prove that a point can be found which is at the same distance from each of the four points$\ldots$

Prove that a point can be found which is at the same distance from each of the four points $\bigg(am_1,\dfrac{a}{m_1}\bigg),\bigg(am_2,\dfrac{a}{m_2}\bigg),\bigg(am_3,\dfrac{a}{m_3}\bigg)$ and ...