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Difference between revisions of "2015 IMO Problems/Problem 3"

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Revision as of 13:48, 1 June 2024

Let $ABC$ be an acute triangle with $AB>AC$. Let $\Gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\Gamma$ such that $\angle HKQ=90^\circ$. Assume that the points $A$, $B$, $C$, $K$, and $Q$ are all different, and lie on $\Gamma$ in this order.

Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

The Actual Problem

Let $ABC$ be an acute triangle with $AB > AC$. Let $\gamma$ be its circumcircle, $H$ its orthocenter, and $F$ the foot of the altitude from $A$. Let $M$ be the midpoint of $BC$. Let $Q$ be the point on $\gamma$ such that $\angle HQA = 90◦$ and let $K$ be the point on $\gamma$ such that $\angle HKQ = 90◦$ . Assume that the points $A$, $B$, $C$, $K$ and $Q$ are all different and lie on $\gamma$ in this order. Prove that the circumcircles of triangles $KQH$ and $FKM$ are tangent to each other.

Solution

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We know that there is a negative inversion which is at $H$ and swaps the nine-point circle and $\gamma$. And this maps:

$A \longleftrightarrow F$. Also, let $M \longleftrightarrow Q`$. Of course $\triangle HFM \sim \triangle HQ'A$ so $\angle HQ'A = 90$. Hence, $Q' = Q$. So:

$M \longleftrightarrow Q$. Let $HA$ and $HQ$ intersect with nine-point circle $T$ and $Q$, respectively. Let's define the point $L$ such that $TNML$ is rectangle. We have found $M \longleftrightarrow Q$ and if we do the same thing, we find:

$L \longleftrightarrow K$. Now, we can say:

$(KQH) \longleftrightarrow ML$ and $(FKM) \longleftrightarrow (ALQ)$. İf we manage to show $ML$ and $(ALQ)$ are tangent, the proof ends.

We can easily say $TN || AQ$ and $AQ = 2.TN$ because $T$ and $N$ are the midpoints of $HA$ and $HQ$, respectively.

Because of the rectangle $TNML$, $TN || ML$ and $TN = ML$.

Hence, $ML || AQ$ and $AQ = 2.ML$ so $L$ is on the perpendecular bisector of $AQ$ and that follows $\triangle ALQ$ is isoceles. And we know that $ML || AQ$, so $ML$ is tangent to $(ALQ)$. We are done. $\blacksquare$

~ EgeSaribas

$Really Important Note:$ This solution is in the "IMO 2015 Solution Notes" which is written by Evan Chen.

There is the link: https://web.evanchen.cc/exams/IMO-2015-notes.pdf

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See Also

2015 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
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