If two intersecting chords of the circle make an equal angle with the diameter passing through their point of contact to the diameter, prove that the chords are equal.

To solve the problem, let us work step by step and prove that the chords are equal. Here’s a detailed explanation:

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**Given:**
1. Two intersecting chords \( AB \) and \( CD \) of a circle intersect at point \( P \).
2. These chords make an equal angle with the diameter passing through \( P \).

**To Prove:**
The lengths of the chords \( AB \) and \( CD \) are equal.

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**Proof:**

1. **Consider the Geometry:**
- Let the diameter passing through the point of intersection \( P \) be \( EF \).
- Denote the angle \( \angle APF = \angle CPF \) (as given in the problem, the chords make equal angles with the diameter).

2. **Establish Symmetry Using Circle Properties:**
- In a circle, the angle subtended by a chord at any point on the circle depends only on the length of the chord.
- Let \( O \) be the center of the circle. The line \( EF \) is the diameter, so \( OP \) is perpendicular to \( AB \) and \( CD \).

3. **Triangles and Symmetry:**
- Focus on the triangles formed by the chords and the diameter:
- In \( \triangle APF \) and \( \triangle CPF \):
- \( \angle APF = \angle CPF \) (given),
- \( PF \) is common,
- \( OP \), being perpendicular, divides the chords symmetrically into two equal halves.

4. **Prove Equal Lengths:**
- Using the Equal Angles:
- If \( \angle APF = \angle CPF \), then the distances \( AP \) and \( CP \) from \( P \) to the ends of the chords \( AB \) and \( CD \) respectively must be the same.
- The symmetry of the circle enforces that the arcs subtended by equal angles are of equal length.
- Consequently, the full chords \( AB \) and \( CD \) must also be equal in length.

5. **Conclusion:**
- Since the chords subtend equal angles with the diameter and the arcs are symmetric, the chords \( AB \) and \( CD \) are equal.

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Thus, we have proved that the chords \( AB \) and \( CD \) are of equal length when they make equal angles with the diameter passing through their point of intersection.