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Through the vertices: O, N, J, K, L and M of the squares supported on the sides AB, BC and AC of the triangle ABC (sides: a, b and c) a conic passes. Another ellipse passes through the points: T, U, Q, P, S and R. Determine the equation of these conics. These points are the solution to the problem of the square inscribed in a triangle.

squareTriangle

Blue
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Wilson Massaro
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  • What have you tried? Where did you encounter your difficulties? – David G. Stork Apr 23 '20 at 19:38
  • If you can find the coordinates of five of the points (which is tedious, but straightforward), then you can use a "simple" $6$-by-$6$ determinant to obtain the equation of the conic containing them. See, for instance, this answer. – Blue Apr 23 '20 at 20:36
  • I crunched some symbols in Mathematica for the first ellipse. The resulting equation is quite ugly, which is to be somewhat expected. For a triangle with convenient coords (such as is illustrated in your figure), the corresponding ellipse will be translated and rotated and ... well ... ugly. Perhaps you could say something about why you need the equation. – Blue Apr 23 '20 at 21:10

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