LX and BC meet at given circle, perpendularity wanted

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LX and BC meet at given circle, perpendularity wanted

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Source: 2019 Nigerian Senior Mathematics Olympiad Round 4 (final) problem 2

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#1 h Sep 9, 2019, 4:53 PM • 2 Y

Y by Adventure10, Mango247

Let $K,L, M$ be the midpoints of $BC,CA,AB$ repectively on a given triangle $ABC$ . Let $\Gamma$ be a circle passing through $B$ and tangent to the circumcircle of $KLM$ , say at $X$ . Suppose that $LX$ and $BC$ meet at $\Gamma$ . Show that $CX$ is perpendicular to $AB$ .

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#2 h Sep 10, 2019, 1:39 PM • 1 Y

Y by Adventure10

Let $H_C$ be the foot of $C$ on line $AB$ . Let $\Omega$ be the nine point circle of triangle $ABC$ . Of course it consists points $K,L,M,H_C$ . Denote $L'$ as the intersection of $LX$ and $BC$ . Circles $\Omega,\Gamma$ are externally tangent, so there exists homothety $\kappa$ centered at $X$ such that $\kappa(\Gamma)=\Omega$ . Because points $L',X,L$ are collinear and $L\in\Omega,L'\in\Gamma$ we have $\kappa(L')=L$ $BC||ML$ so segments $L'B,ML$ are parallel chords of circles $\Gamma,\Omega$ respectively. Thus $\kappa(L'B)=ML$ Both reached equalities lead to $\kappa(B)=M$ .
Hence $B,X,M$ are collinear $X\in AB$ . From the definition of the point $X\in \Omega$ . Finally $X=H_C$ .

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#3 h Sep 10, 2019, 4:20 PM • 2 Y

Y by Adventure10, Mango247

Is it Externally tangent or internally tangent

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Reason: Incomplete

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#4 h Sep 10, 2019, 5:02 PM • 1 Y

Y by Adventure10

Internally when $\angle ABC>90^\circ$ and externally when $\angle ABC<90^\circ$ . When $\angle ABC=90^\circ$ the circle $\Gamma$ is a point $X=B$ .

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#5 h Sep 10, 2019, 5:13 PM • 2 Y

Y by Adventure10, Mango247

I think this case is internally because the nine point circle Is inside the Circle $\Gamma$

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Reason: Latex problem

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#6 h Sep 10, 2019, 5:54 PM • 1 Y

Y by Adventure10

The problem doesn't need to mention whether it's internally or externally tangent case, because that is dependent only on the position of $B$ with respect to the nine point circle. My solution is correct for all cases and so is the problem.

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