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2020 CAMO Problems/Problem 3

Revision as of 05:47, 29 September 2025 by Alexanderchew (talk | contribs) (added new solution)

Problem 3

Let $ABC$ be a triangle with incircle $\omega$, and let $\omega$ touch $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Point $M$ is the midpoint of $\overline{EF}$, and $T$ is the point on $\omega$ such that $\overline{DT}$ is a diameter. Line $MT$ meets the line through $A$ parallel to $\overline{BC}$ at $P$ and $\omega$ again at $Q$. Lines $DF$ and $DE$ intersect line $AP$ at $X$ and $Y$ respectively. Prove that the circumcircles of $\triangle APQ$ and $\triangle DXY$ are tangent.

Solution

Let $R$ be the intersection of $AT$ with $\omega$ other than $T$.

Claim 1: $R$ is the $D$-queue point of $DXY$. In particular, $R$ lies on $(DXY)$. Proof: First, we claim that $X$ and $Y$ lie on $TE$ and $TF$ respectively. Let $X_0=TE\cap DF$ and $Y_0=TF\cap DE$. By Brokard, $X_0Y_0$ is the polar of $EF\cap DT$ which is the line through $A$ parallel to $BC$, so it coincides with the line $XY$ which gives $X\equiv X_0$ and $Y\equiv Y_0$. The fact that $\measuredangle TED = \measuredangle TFD = 90^{\circ}$ is enough to imply that $T$ is the orthocenter. Now, \begin{align*} (X, Y; A, \infty_{XY}) &\overset{T}{=} (E, F; R, T) \\ &= -1 \end{align*} so $A$ is the midpoint of $XY$, which gives the claim.

Denote $f_1$ as the inversion centered at $T$ swapping $(DXY)$ with its nine-point circle $(AEF)$, and denote $f_2$ as the inversion centered at $A$ fixing $\omega$.

Claim 2: $R$ lies on $(APQ)$. Proof: Note that $f_1(Q)=P$ which follows from the fact that $f_1(\omega)=XY$, so we are done with this claim by Power of a Point Theorem.

Finally, we return to prove that $R$ is the tangency point of $(DXY)$ and $(APQ)$. Inverting at $f_1$ and $f_2$ in that order, it suffices to show that \[f_2(f_1(DXY))=EF\] and \[f_2(f_1(DXY))=H'T\] are parallel, where $H'$ is defined to be the intersection of $AQ$ and $\omega$ other than $Q$.

But note that $(Q, H'; E, F)=-1$ and $Q$ is the $D$-queue point of $DEF$, so it is well known that $H'$ is the point on $\omega$ such that $DH'\perp EF$. The parallelism then follows from here.

See also

2020 CAMO (Problems • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6
All CAMO Problems and Solutions

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