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The image shows a geometric configuration with a circle and points labeled \( A \), \( B \), \( C \), and \( O \) (the center of the circle). The goal is to prove the following relation:

\[
AT^2 = AB \times AC
\]

This is a standard result in geometry related to tangents and secants drawn from an external point to a circle. Here, \( AT \) represents the length of the tangent drawn from point \( A \) to the circle, and \( AB \) and \( AC \) are the lengths of the segments created by a secant from \( A \) intersecting the circle at points \( B \) and \( C \).

### Proof Outline:

1. **Power of a Point Theorem**:
   The **Power of a Point Theorem** states that for a point \( A \) outside a circle, the square of the length of the tangent from \( A \) to the circle is equal to the product of the lengths of the secant segments.
   
   Mathematically, this theorem gives us:
   \[
   AT^2 = AB \times AC
   \]
   where:
   - \( AT \) is the length of the tangent from point \( A \) to the circle,
   - \( AB \) is the distance from \( A \) to \( B \),
   - \( AC \) is the distance from \( A \) to \( C \), and point \( B \) lies on the secant between \( A \) and \( C \).

2. **Application to the Given Figure**:
   - \( A \) is the external point.
   - \( AT \) is the tangent from \( A \) to the circle.
   - \( B \) and \( C \) are points where the secant from \( A \) intersects the circle.

   According to the Power of a Point Theorem, this configuration directly satisfies the relation \( AT^2 = AB \times AC \).

Thus, the proof follows from the Power of a Point Theorem.

Would you like more details, or do you have any questions?

### Related Questions:
1. What are the different parts of a circle related to tangents and secants?
2. How does the Power of a Point Theorem apply in other geometric configurations?
3. Can the Power of a Point Theorem be generalized to other shapes?
4. How can we derive the equation of a tangent to a circle in analytic geometry?
5. What are other important theorems involving circles, tangents, and chords?

**Tip**: Always remember the Power of a Point Theorem for configurations involving tangents and secants, as it simplifies many circle-related problems.

To prove: AT^2 = AB × AC, based on the given geometric configuration involving a circle and points A, B, C, and a tangent from point A.

Explore the geometric proof that AT^2 = AB × AC, a standard result in circle geometry using the Power of a Point Theorem. This problem involves tangents and secants drawn from a common external point. Suitable for students in Grades 9-12.

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