In the diagram, chords AB, BC and AC are drawn in the circle with centre O - NSC Mathematics - Question 8 - 2024 - Paper 2

To prove that the angle BCE is equal to angle Â, we can follow these steps:

1. Identify Key Angles: The angle DCE, which is formed by the tangent at point C and the line segment CE, is equal to the inscribed angle BAC since they subtend the same arc AC. Thus, we have:

   $DCE = BAC$

2. Using the Inscribed Angle Theorem: According to this theorem, the angle subtended by an arc at the center of the circle is double that of the angle subtended at any point on the circumference. Considering triangle ABC, we see that:

   $BCE = 90° - DCE$

3. Relating the Angles: Since angle BAC is subtended by arc BC, we apply:

   $BCE + BAC = 90°$

4. Final Derivation: This means that angles BCE and Â, which are both subtended by the chord AC, are equal based on the above relationships:

   $BCE = Â$

Thus, we have proven that the angle between the tangent and the chord at the point of contact is equal to the angle in the alternate segment.