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$MO \cap (O) = E, F$. $MAB$ and $MN$ are respectively a secant and a tangent of $(O)$ at $N$ ($ME < MA < MN < MB < MF$).The tangent of $(O)$ at $E$ intersects semicircle diameter $MF$ at $K$. $NO \cap FK = T$. Prove that the circumcentre of $\triangle ABT$, $\triangle EFT$ and the midpoint of $KT$ collinear.

I am trying to prove the following claims leading to the solution of the problem.

  • $KN \perp MT$

  • $MT \perp PQ$ (or $MT$ is the tangent of two externally circumcircles of $\triangle ABT$ and $\triangle EFT$)

  • $RS \parallel KN$ (or $S$ is the midpoint of $NT$ with $S = NT \cap PQ$, $P$ and $Q$ are respectively the circumcentre of $\triangle ABT$ and $\triangle EFT$)

This is supposed to be an easy problem. Why couldn't I solve it?

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Consider three circles: the circumcircle of $ATB$, the circumcircle of $ETF$ and the circle with diameter $KT$. All of them pass through $T$. To show that their centers lie on a line, we will show that they have common radical axis. $T$ has the same power $0$ with respect to the circles.

Now, let's prove that M has the same power with repect to the circles. Firstly, $MA\cdot MB=ME\cdot MF$. Secondly, the triangle $MKF$ is right, and $KE$ is its height to hypotenuse, so $ME\cdot MF=MK^2$

That's all.

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  • $\begingroup$ @LêThànhĐạt well, the proof of the radical axis theorem uses coordinates, but once you've proven that, there are no more coordinates involved. I wouldn't say that this solution uses coordinates. $\endgroup$ – liaombro May 7 at 15:57
  • $\begingroup$ @LêThànhĐạt actually, you can prove the radical axis theorem using only Pythagoras theorem. Is Pythagoras coordinate geometry? $\endgroup$ – liaombro May 7 at 16:08
  • $\begingroup$ It's a shame that radical axis is not something I have learnt at school (but something I self-study and came by). Because this is my homework and my teacher wouldn't accept my proof (although they know perfectly what radical axis), could you say how can I prove the property of radical axis that you used in your proof without the definition of radical axis, please? $\endgroup$ – Lê Thành Đạt May 7 at 16:09

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