Let l and m be distinct lines intersecting at P. let there be two points x and y on l and m respectively such that neither x nor y is equal to P. Let k be the circle through the points P, x and y (P is on the circumference). Construct tangents to X and Y. Prove that, if l and m are perpendicular, the tangents to x and y are parallel.

To prove that the tangents to the points \( X \) and \( Y \) on the circle \( k \), with respect to the lines \( l \) and \( m \), are parallel given that \( l \) and \( m \) are perpendicular, we will use the following approach:

### Step-by-Step Construction and Proof:

1. **Given Information:**
   - Let \( l \) and \( m \) be two distinct, perpendicular lines intersecting at a point \( P \).
   - There are two points \( X \) on \( l \) and \( Y \) on \( m \), neither of which are equal to \( P \).
   - A circle \( k \) passes through the points \( P \), \( X \), and \( Y \).
   - We aim to show that the tangents to the circle at the points \( X \) and \( Y \) are parallel.

2. **Circle Geometry and Tangents:**
   - Let the circle \( k \) be centered at \( O \), with \( P \), \( X \), and \( Y \) on its circumference.
   - The tangent to a circle at any point is perpendicular to the radius at that point. Therefore:
     - The tangent to the circle at \( X \) is perpendicular to the radius \( OX \).
     - The tangent to the circle at \( Y \) is perpendicular to the radius \( OY \).

3. **Orthogonality and Perpendicularity:**
   - Since \( X \) lies on line \( l \) and \( Y \) lies on line \( m \), both of which intersect perpendicularly at \( P \), it follows that the angle \( \angle XPY = 90^\circ \).
   - By the Inscribed Angle Theorem (or Thales' Theorem), since \( \angle XPY = 90^\circ \), the arc subtended by \( X \) and \( Y \) with respect to the circle is a semicircle. This means that \( O \), the center of the circle, must lie on the line perpendicular to the line segment \( XY \) at its midpoint.

4. **Symmetry of the Tangents:**
   - The symmetry of the circle due to the perpendicular intersection of lines \( l \) and \( m \) implies that the tangents to the circle at \( X \) and \( Y \) are symmetric with respect to the line \( XY \).
   - More precisely, the tangent at \( X \) is perpendicular to the radius \( OX \), and the tangent at \( Y \) is perpendicular to the radius \( OY \). Since the radii \( OX \) and \( OY \) are symmetric about the line \( XY \), the tangents to \( X \) and \( Y \) must also be symmetric about this line.

5. **Conclusion:**
   - Therefore, the tangents to the circle at \( X \) and \( Y \) are parallel to each other, as the symmetry of the setup (with \( l \) and \( m \) being perpendicular) forces them to have the same slope.

Thus, if the lines \( l \) and \( m \) are perpendicular, the tangents to the points \( X \) and \( Y \) on the circle are parallel.

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Would you like a more detailed breakdown of any part of the proof?

### Related Questions:
1. How do the properties of tangents to a circle relate to perpendicular lines?
2. What is the role of symmetry in geometry, especially with circles?
3. How does the Inscribed Angle Theorem help in this proof?
4. Can this result be extended to other geometrical configurations involving circles?
5. What are the conditions for two tangents to a circle to be parallel?

### Tip:
In problems involving tangents to circles, always remember that the tangent is perpendicular to the radius at the point of contact. This fact is crucial in many circle geometry proofs.

Let l and m be distinct lines intersecting at P. Let there be two points x and y on l and m respectively such that neither x nor y is equal to P. Let k be the circle through the points P, x, and y (P is on the circumference). Construct tangents to X and Y. Prove that, if l and m are perpendicular, the tangents to x and y are parallel.

Explore a detailed geometric proof showing that the tangents to points X and Y on a circle are parallel when the lines l and m, which intersect at point P, are perpendicular. The problem involves key concepts like tangents, the Inscribed Angle Theorem, and Thales' Theorem. Ideal for high school geometry students in Grades 9-12.

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