https://file.aiquickdraw.com/mathbot/file/1725520668568lyts8zw2.jpg

I see you have uploaded an image of a geometry problem. Let me analyze the content and provide an explanation.

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The diagram shows a circle with a triangle inscribed in it. The circle has center \(O\), and the points \(C\), \(E\), and \(F\) appear to form a triangle, with point \(F\) outside the circle and points \(C\) and \(E\) on the circumference of the circle.

The accompanying text appears to be asking for the following:

- To find the size of \(\angle FCE\).
- To find the size of \(\angle CFE\), which involves the use of properties of angles at the center and circumference of a circle, likely referencing that the angle at the center is twice the angle at the circumference.
- The next part involves constructing a tangent and understanding a right angle formed by the tangent and the radius at the point of contact.

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I will proceed by giving you a general outline for solving the angle-based parts of this geometry problem based on common theorems.

### Key Concepts
1. **Angle at the Circumference Theorem**: An angle subtended by a chord at the circumference of a circle is half the angle subtended at the center.
   - \(\angle FCE\) can be determined by looking at how \(\angle COE\) (the central angle) subtends the arc \(CE\).
  
2. **Tangent-Radius Theorem**: A tangent to a circle is perpendicular to the radius at the point of tangency.
   - This helps with understanding right angles formed with tangents.

Would you like detailed calculations or explanations for each part? Additionally, let me know if you have any questions or need further clarifications!

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### 5 Related Questions:
1. How do you calculate angles subtended by the same arc in a circle?
2. What is the relationship between the central angle and the inscribed angle subtended by the same arc?
3. How does the tangent to a circle relate to the radius at the point of contact?
4. How can you determine the size of an exterior angle in a triangle?
5. What is the significance of cyclic quadrilaterals in circle geometry?

### Tip:
Always look for key circle theorems (like the angle subtended by the same arc or the tangent-radius property) to solve problems involving angles in circles.

In the given diagram, find the size of ∠FCE and the size of ∠CFE, given that F is outside the circle, C and E are on the circle's circumference, and O is the center.

Solve this challenging geometry problem involving angles in a circle, including the Angle at the Circumference Theorem and the Tangent-Radius Theorem. Perfect for high school students in Grades 9-11.

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