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$ABC$ is an acute triangle which is inscribed circle $(O)$. A line through $O$ cut $AB,AC$ at $M,N$. Prove that circle with diameter $BN$, circle with diameter $CM$ and Euler circle of traingle $ABC$ concur. enter image description here

I don't know the way to solve this. I think it is impossible to use basic Euclid geometry theorem. There are lots of points on circles, where we can use pascal theorem, but how to use? Please solve me the added geometry factor to draw. Thanks.

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  • $\begingroup$ There is a simple approach: apply a circle inversion with respect to the circle with diameter $BC$. $\endgroup$ – Jack D'Aurizio Dec 19 '18 at 20:59

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