Problem 1535

1. Identify the given elements and their properties:
- $A, B, C, D$ are consecutive vertices of a regular $7$-gon.
- $AL$ and $AM$ are tangents to the circle centered at $C$ with radius $CB$.
- $N$ is the intersection point of $AC$ and $BD$.

2. Establish the cyclic nature of the polygons:
- Since $A, B, C, D$ are vertices of a regular $7$-gon, they lie on a common circle (the circumcircle of the $7$-gon).
- Points $L, B, M, D$ lie on a circle with center $C$ and radius $CB$ because $L$ and $M$ are points where tangents from $A$ touch the circle centered at $C$.

3. **Prove that $ALCM$ is cyclic:**
- Since $AL$ and $AM$ are tangents to the circle centered at $C$, we have $CL \perp AL$ and $CM \perp AM$.
- This implies that $\angle CLA = \angle CMA = 90^\circ$.
- Therefore, quadrilateral $ALCM$ is cyclic because opposite angles sum to $180^\circ$.

4. Apply the Radical Axis Theorem:
- The Radical Axis Theorem states that for three circles, the radical axes of each pair of circles are concurrent.
- Consider the three circles:
- The circumcircle of $ABCD$.
- The circle with center $C$ and radius $CB$ (containing $L, B, M, D$).
- The circle containing $ALCM$.
- The radical axis of the circumcircle of $ABCD$ and the circle containing $L, B, M, D$ is line $BD$.
- The radical axis of the circumcircle of $ABCD$ and the circle containing $ALCM$ is line $AC$.
- The radical axis of the circle containing $L, B, M, D$ and the circle containing $ALCM$ is line $LM$.

5. **Show concurrency of $AC$, $BD$, and $LM$:**
- By the Radical Axis Theorem, the lines $AC$, $BD$, and $LM$ are concurrent.
- Since $N$ is the intersection point of $AC$ and $BD$, it must also lie on $LM$.

6. **Conclude that $L, M, N$ are collinear:**
- Since $N$ lies on $LM$, points $L, M, and N$ are collinear.

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