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In a triangle ABC Let a point X be the point of tangency of the excircle opposite A with side BC. I am trying to prove the point X cannot cannot lie on the nine-point circle of triangle ABC. I proved that there could not be three intersections across side BC, but how can I prove that X can't possibly be a point on side BC that coincides with the midpoint or the other point of intersection of the nine-point circle?

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Let the triangle be isosceles and then use Feuerbach's Theorem and Feuerbach's Point to prove the conjecture.

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