https://file.aiquickdraw.com/mathbot/file/1727365263160pboz3sps.jpg

The image contains two geometry problems, both involving proofs:

1. **First Problem**:
   - **Given**: Two right-angled triangles, \( \Delta ABC \) and \( \Delta DBC \), share the same hypotenuse \( BC \). The lines \( BD \) and \( AC \) intersect at a point \( P \).
   - **To Prove**: 
     \[
     \frac{AP}{PC} = \frac{BP}{PD}
     \]

   This problem can be solved using properties of similar triangles. Since \( \Delta ABC \) and \( \Delta DBC \) share the same right angle at \( B \) and have the same hypotenuse \( BC \), the triangles are similar by AA similarity (Angle-Angle). The ratio of corresponding sides in similar triangles can be used to establish the required proportion.

2. **Second Problem**:
   - **Given**: A parallelogram \( ABCD \) is drawn. In the figure, \( AB = CF \), \( BE = FB \), and \( AE \) is a diagonal or segment.
   - **To Prove**:
     \[
     AB = CF
     \]
   
   This is straightforward since \( ABCD \) is a parallelogram, and one of the properties of parallelograms is that opposite sides are equal. So, by using the properties of parallelograms and segment relations within the diagram, you can establish the equality.

### Detailed Solutions:

1. **First Proof**: 
   - Identify the similar triangles: 
     - \( \Delta ABP \sim \Delta CDP \)
     - Since they are similar, corresponding sides are proportional:
       \[
       \frac{AP}{PC} = \frac{BP}{PD}
       \]
   - This directly proves the required equality.

2. **Second Proof**:
   - Use the properties of parallelograms: opposite sides are equal, and diagonal properties may help confirm segment equivalences within the figure.

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Would you like to go through the detailed steps for these proofs or have any questions? Here are some related questions for further practice:

1. What is the AA similarity criterion for triangles?
2. How do the properties of parallelograms help in geometric proofs?
3. Can you prove the ratio \( \frac{AP}{PC} \) using coordinate geometry?
4. What other methods could be used to prove the equality of opposite sides in a parallelogram?
5. How can you apply the midpoint theorem in the second problem?

**Tip**: In geometry problems involving ratios, always look for potential similar triangles or proportional relationships between segments.

ΔABC and ΔDBC are two right-angled triangles on the same hypotenuse. BC, BD intersects AC at point P. Prove that AP/PC = BP/PD. In the second figure, ABCD is a parallelogram. Prove that AB = CF.

Explore the geometric proofs involving the similarity of triangles and properties of parallelograms. This includes proving proportional relationships between triangle sides and establishing segment equivalencies using parallelogram properties. Ideal for students in Grades 9-10.

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