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Technically, yes. But, the equations involved are complicated to the point that it would be a fraction of the difficulty of converting. Also, the equations are essentially the Cartesian equations with the conversions built in, so you might as well convert them to start with.

However, if you insist on not converting, write out the entire process with all 4 points of interest in Cartesian coordinates. From beginning to end. Find the final equations needed and insert the conversion factors and simplify from there. To the best of my knowledge (and I did quite a bit of digging) there isn't a simply way of doing it. - Sorry.

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Q: Given three points in polar notation what are the polar coordinates of a point equidistant from the three given points and can this point be found without converting to Cartesian coordinates?
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