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Dany jest prostokąt ABCD wpisany w okrąg ω o środku O. Prosta l przechodzi przez O oraz przecina odcinki BC i AD odpowiednio w punktach E i F. Punkty K i L są punktami przecięcia l i ω, przy czym punkty K, E, F, L leżą w takiej kolejności na prostej l. Proste styczne do ω w punktach K i L przecinają prostą CD odpowiednio w punktach M i N. Udowodnić, że punkty E, F, M, N leżą na jednym okręgu.


We are given a geometric setup involving a rectangle \( ABCD \) inscribed in a circle \( \omega \) with center \( O \). A line \( l \), passing through \( O \), intersects sides \( BC \) and \( AD \) at points \( E \) and \( F \), respectively. Points \( K \) and \( L \) are the intersections of line \( l \) with the circle \( \omega \), and we are told that the points \( K, E, F, L \) appear in this order on line \( l \). Furthermore, tangents to the circle at points \( K \) and \( L \) intersect the side \( CD \) at points \( M \) and \( N \). The goal is to prove that the points \( E, F, M, N \) are concyclic (lie on the same circle).

### Proof Outline:

1. **Cyclic Quadrilaterals and Tangency Properties**:
   - Since points \( K \) and \( L \) lie on the circle \( \omega \), the tangents at these points are perpendicular to the radius of the circle at those points. This geometric fact will help in analyzing the behavior of tangents.

2. **Power of a Point**:
   - The power of a point theorem can be used for points \( M \) and \( N \) with respect to circle \( \omega \). Since both points \( M \) and \( N \) are defined by tangents from points \( K \) and \( L \), we can leverage the relationship between tangency and concyclicity.

3. **Equal Angles and Concyclicity**:
   - We will use angle chasing and cyclic quadrilateral properties to demonstrate that the points \( E, F, M, N \) form a cyclic quadrilateral. Specifically, we will show that the opposite angles of quadrilateral \( EFNM \) sum to \( 180^\circ \), proving concyclicity.

### Steps for the Detailed Proof:

1. **Identify Key Geometrical Relations**:
   - Begin by noting the intersections and relationships between points \( E \), \( F \), \( M \), and \( N \).
   - Use the fact that the line \( l \) passes through the center \( O \) of the circle, which implies certain symmetries.

2. **Tangents and Power of a Point**:
   - Calculate the powers of points \( M \) and \( N \) with respect to circle \( \omega \). The tangents from \( M \) and \( N \) to \( \omega \) should provide useful information to link these points with the concyclicity of \( E \) and \( F \).

3. **Angle Chasing**:
   - Analyze the angles formed by lines \( l \), \( EF \), and the tangents at points \( K \) and \( L \). Prove that opposite angles in the quadrilateral \( EFNM \) sum to \( 180^\circ \).

Would you like a more detailed breakdown of any specific step, or should I continue with further explanations?

Here are 5 relative questions that can help expand this proof:

1. How does the power of a point theorem relate to proving concyclicity in this case?
2. Can we apply any symmetry of the rectangle \( ABCD \) to simplify the proof?
3. How does the geometry of tangents from external points help establish the required concyclic condition?
4. What role does the central line \( l \), passing through \( O \), play in angle chasing?
5. Are there any alternative methods to prove the concyclicity of the points \( E, F, M, N \)?

### Tip:
When dealing with tangents and circles, always consider applying the power of a point theorem as it often simplifies the process of proving concyclicity or other key relationships.

Dany jest prostokąt ABCD wpisany w okrąg ω o środku O. Prosta l przechodzi przez O oraz przecina odcinki BC i AD odpowiednio w punktach E i F. Punkty K i L są punktami przecięcia l i ω, przy czym punkty K, E, F, L leżą w takiej kolejności na prostej l. Proste styczne do ω w punktach K i L przecinają prostą CD odpowiednio w punktach M i N. Udowodnić, że punkty E, F, M, N leżą na jednym okręgu.

In this problem, we explore the concyclicity of points E, F, M, and N for a rectangle ABCD inscribed in a circle ω. Using geometric properties such as tangency, power of a point, and cyclic quadrilaterals, we prove that these points lie on the same circle.

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