AC and BC are two equal chords of a circle. BA is produced to any point P and CP, when joined cuts the circle at T. Then

Options

A. `CT : TP = AB : CA`
B. `CT : TP = CA : AB`
C. `CT : CB = CA : CP`
D. `CT : CB = CP : CA`

Answer

To solve the problem step by step, we will analyze the given information about the circle and the chords AC and BC, and then derive the required relationships.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- We have a circle with two equal chords AC and BC.
- BA is extended to a point P.
- The line CP intersects the circle at point T.

2. **Draw the Diagram**:
- Draw a circle and mark the points A, B, C such that AC = BC.
- Extend BA to point P.
- Draw line CP and mark the point where it intersects the circle as T.

3. **Use Properties of Equal Chords**:
- Since AC and BC are equal chords, the angles opposite to these chords in triangle ABC will also be equal. Therefore, we have:
\[ \angle BAC = \angle ABC \]
- Let these angles be denoted as \( x \).

4. **Analyze Triangle PAC**:
- In triangle PAC, we can find angle PAC:
\[ \angle PAC = 180^\circ - x \quad \text{(since angles on a straight line sum to 180 degrees)} \]

5. **Cyclic Quadrilateral Properties**:
- The quadrilateral ABCT is cyclic because points A, B, C, and T lie on the circle.
- For cyclic quadrilaterals, the sum of opposite angles is 180 degrees:
\[ \angle ABC + \angle ATC = 180^\circ \]
- Since \( \angle ABC = x \), we can write:
\[ \angle ATC = 180^\circ - x \]

6. **Equate Angles**:
- From the previous steps, we have:
\[ \angle ATC = 180^\circ - x \quad \text{and} \quad \angle PAC = 180^\circ - x \]
- Therefore, we can conclude:
\[ \angle ATC = \angle PAC \]

7. **Establish Similarity of Triangles**:
- Since we have two angles equal (\( \angle ATC = \angle PAC \) and \( \angle ACP = \angle ACT \)), by the Angle-Angle (AA) criterion, triangles ATC and PAC are similar:
\[ \triangle ATC \sim \triangle PAC \]

8. **Set Up Proportionality**:
- From the similarity of triangles, we can write the ratio of corresponding sides:
\[ \frac{AC}{PC} = \frac{AT}{PA} \]
- Since AC = BC, we can also express this as:
\[ \frac{AC}{PC} = \frac{CT}{CB} \]

9. **Final Result**:
- We conclude that:
\[ \frac{CA}{CP} = \frac{CT}{CB} \]
- This corresponds to the given options, confirming that the correct answer is option 3.