Amazing Triangle Problem

Here is another simply amazing problem from Five Hundred Mathematical Challenges:

Problem 154. Show that three solutions, (x1,.y1), (x2,.y2), (x3, y3), of the four solutions of the simultaneous equations
____________(x – h)² + (y – k)² = 4(h² + k²)
______________________xy = hk
are vertices of an equilateral triangle. Give a geometrical interpretation.”

Again, I don’t see how anyone could have discovered this property involving a circle, a hyperbola, and an equilateral triangle. It seems plausible when h.=.k, but it is not at all obvious for h..k. For some reason, I had difficulty getting a start on a solution, until the obvious approach dawned on me. I don’t know why it took me so long.

See the Amazing Triangle Problem.

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