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Analytic Geometry

Coordinates, mass points, and even some complex numbers.

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Points \(A, B, \) and \(C\) lie at \((0,0), (3,0)\), and \((0,4)\) respectively. What is the perimeter of \(\triangle ABC\)?

A circle contains points \((0,4)\) and \((3,0)\). What is the equation of the line which contains all possible centers of this circle?

Let \(ABC\) be a triangle, and let \(D, E, F\) be the midpoints of \(BC, AC, AB\) respectively. Let \(G\) be the intersection of \(AD, BE,\) and \(CF\). Find \(\frac{AG}{GD} + \frac{BG}{GE} + \frac{CG}{GF}\).

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