Suppose ABC is an acute angled triangle. Let O be the center of the circumcircle of ABC. The line AC intersects the circumcircle of triangle ABO at T (where T is different to the point A). Prove that the lines OT and BC are perpendicular.

We are given an acute-angled triangle \( ABC \) with \( O \) as the circumcenter of \( \triangle ABC \). The line \( AC \) intersects the circumcircle of \( \triangle ABO \) at a point \( T \) different from \( A \). We need to prove that the lines \( OT \) and \( BC \) are perpendicular.

### Step 1: Understanding the configuration
Let \( \Gamma \) be the circumcircle of \( \triangle ABC \), and let \( \Gamma' \) be the circumcircle of \( \triangle ABO \). By construction, \( T \) lies on \( \Gamma' \), and \( OT \) is the line from the circumcenter \( O \) of \( \Gamma \) to \( T \).

### Step 2: Using the Radical Axis Theorem
We will use the Radical Axis Theorem, which states that if three circles are given, the radical axes of each pair of circles are either concurrent or parallel.

Consider the three circles:
1. The circumcircle \( \Gamma \) of \( \triangle ABC \),
2. The circumcircle \( \Gamma' \) of \( \triangle ABO \),
3. The circle at infinity (the line at infinity in the context of radical axes).

The radical axis of \( \Gamma \) and \( \Gamma' \) is the line \( BC \), as this is the radical axis of the two circles with points \( B \) and \( C \) being common to both circles.

The radical axis of \( \Gamma \) and the circle at infinity is the line \( OT \), because \( O \) is the center of \( \Gamma \) and \( T \) is a point where the circle \( \Gamma \) and another point (at infinity) intersect.

According to the Radical Axis Theorem, the radical axis \( OT \) (of \( \Gamma \) and the circle at infinity) is perpendicular to the radical axis \( BC \) (of \( \Gamma \) and \( \Gamma' \)).

### Step 3: Conclusion
Thus, the lines \( OT \) and \( BC \) are perpendicular to each other, as required.

This completes the proof.

Would you like further details or have any questions?

Here are some related questions for further exploration:
1. What is the Radical Axis of two circles?
2. How does the Radical Axis Theorem apply to three given circles?
3. Can this proof be extended to obtuse-angled triangles? Why or why not?
4. What are the properties of the circumcenter in acute and obtuse triangles?
5. How does the configuration change if \( T \) lies on a different line than \( AC \)?

**Tip:** When dealing with problems involving circles and their properties, consider the Radical Axis Theorem as a powerful tool for finding relationships between lines and circles.

Learn how to prove that lines OT and BC are perpendicular in an acute-angled triangle ABC with detailed step-by-step reasoning. This problem involves geometric concepts such as the circumcircle, triangle intersections, and application of the Radical Axis Theorem.

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