Problem 1673

1. Identify the given elements and their properties:
- $ABCD$ is a parallelogram with $\angle DAB < 90^\circ$.
- $E$ is a point on the line $BC$ such that $AE = AB$.
- $F$ is a point on the line $CD$ such that $AF = AD$.
- The circumcircle of $\triangle CEF$ intersects the line $AE$ again at $P$ and the line $AF$ again at $Q$.
- $X$ is the reflection of $P$ over the line $DE$.
- $Y$ is the reflection of $Q$ over the line $BF$.

2. **Prove that $ADPF$ and $ABQE$ are isosceles trapezoids:**
- Since $AE = AB$ and $AF = AD$, triangles $ABE$ and $ADF$ are isosceles.
- In $\triangle ADF$, since $AF = AD$, $\angle AFD = \angle ADF$.
- In $\triangle ABE$, since $AE = AB$, $\angle AEB = \angle ABE$.
- Since $P$ lies on the circumcircle of $\triangle CEF$ and intersects $AE$ again, $\angle AEP = \angle AFP$.
- Similarly, since $Q$ lies on the circumcircle of $\triangle CEF$ and intersects $AF$ again, $\angle AFQ = \angle AEQ$.

3. **Show that the circumcircles of $ADPF$ and $ABQE$ meet at $A$ and another point $Z$:**
- Let the circumcircles of $ADPF$ and $ABQE$ intersect at $A$ and another point $Z$.
- Since $ADPF$ is an isosceles trapezoid, $\angle ADP = \angle AFP$.
- Since $ABQE$ is an isosceles trapezoid, $\angle ABQ = \angle AEQ$.
- Therefore, $\angle AZE = \angle AQB = \angle AFD$.

4. **Prove that $E, Z, D$ and $F, Z, B$ are collinear:**
- Since $\angle AZE = \angle AFD$, $E, Z, D$ must be collinear.
- Similarly, since $\angle AZF = \angle AEB$, $F, Z, B$ must be collinear.

5. **Determine the positions of $X$ and $Y$:**
- Let the perpendicular from $P$ to $DE$ intersect $AZ$ at $X'$.
- Since $\angle DZX = \angle AFD = \angle PZD$ and $AF = AD = DP$, $X = X'$.
- Similarly, let the perpendicular from $Q$ to $BF$ intersect $AZ$ at $Y'$.
- Since $\angle BZY = \angle AEB = \angle QZB$ and $AE = AB = BQ$, $Y = Y'$.

6. **Conclude that $A, X, Y$ lie on the same line:**
- Since $X$ and $Y$ are reflections of $P$ and $Q$ over $DE$ and $BF$ respectively, and both lie on $AZ$, $A, X, Y$ are collinear.

$\blacksquare$