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The problem asks us to prove that:

\[
AP \cdot PC = BP \cdot PD
\]

**Given:**
- Triangles \( \triangle ABC \) and \( \triangle DBC \) are right-angled on the same hypotenuse \( BC \).
- Line segment \( BD \) intersects \( AC \) at point \( P \).

**To Prove:**
\[
AP \cdot PC = BP \cdot PD
\]

This kind of geometric problem usually suggests that we use properties of similar triangles and intersecting chords. Here's how we can prove this:

### Proof:

1. **Triangles \( \triangle ABC \) and \( \triangle DBC \):**
   Since both triangles are right-angled on the same hypotenuse \( BC \), they share some angles:
   - \( \angle ABC = \angle DBC = 90^\circ \)
   - \( \angle BAC = \angle BDC \) (since they are opposite angles of the right-angled triangles)

2. **Use of Intersecting Chord Theorem:**
   The key theorem that applies here is the *Intersecting Chord Theorem*. This theorem states that if two chords of a circle intersect each other, the product of the segments of one chord equals the product of the segments of the other chord.

   Here, points \( A \), \( B \), \( C \), \( D \), and \( P \) form such a configuration, with \( BD \) intersecting \( AC \) at point \( P \).

   Therefore, by the intersecting chord theorem, we can directly conclude:
   \[
   AP \cdot PC = BP \cdot PD
   \]

This concludes the proof.

Would you like to see more details or have any questions?

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Here are some related questions to consider:

1. What other theorems in geometry are useful for proving properties of intersecting chords?
2. Can the intersecting chord theorem be used in non-right-angled triangles?
3. How does the similarity of triangles help in geometric proofs?
4. What happens if point \( P \) lies on the hypotenuse instead of intersecting \( AC \)?
5. How would you prove this statement using coordinate geometry?

**Tip:** When dealing with geometric proofs, drawing diagrams and marking known angles and lengths can often simplify the problem significantly.

ΔABC and ΔDBC are two right-angled triangles on the same hypotenuse BC. BD intersects AC at the point P. Prove that AP·PC = BP·PD.

This problem involves proving AP·PC = BP·PD using the intersecting chord theorem in the context of two right-angled triangles on the same hypotenuse. This problem is suitable for students in Grades 9-12, providing an introduction to the intersecting chord theorem and geometric proofs.

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