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The problem in the image involves a circle with center \( O \) and a triangle \( ABC \). Points \( A \), \( B \), and \( C \) lie on the circle, and the center \( O \) is given with a radius of 8 cm. There is also a point \( E \) on the circle such that:

- \( AE = 10 \) cm
- \( OE = 6 \) cm
- Triangle \( AOE \) is similar to triangle \( ABC \)
  
We are tasked with finding the length of \( BC \).

### Solution:
1. **Similarity of Triangles**:
   Given that triangles \( AOE \) and \( ABC \) are similar, the ratio of corresponding sides of the triangles is the same. Specifically, this means:
   \[
   \frac{AO}{AB} = \frac{OE}{BC}
   \]

2. **Using the Given Values**:
   We know the following:
   - \( AO = 8 \) cm (radius of the circle)
   - \( OE = 6 \) cm
   - \( AE = 10 \) cm
   Now, let the length of \( BC \) be \( x \).

3. **Setting up the Proportion**:
   Using the similarity, we can set up the proportion:
   \[
   \frac{AO}{AB} = \frac{OE}{BC}
   \]
   Substituting the known values:
   \[
   \frac{8}{AB} = \frac{6}{x}
   \]

4. **Finding \( AB \)**:
   To find \( AB \), we use the fact that \( \triangle AOE \) is a right triangle. Using the Pythagorean theorem in \( \triangle AOE \):
   \[
   AO^2 = AE^2 - OE^2
   \]
   Substituting values:
   \[
   8^2 = 10^2 - 6^2
   \]
   \[
   64 = 100 - 36
   \]
   \[
   64 = 64
   \]
   So, \( AB = 8 \) cm.

5. **Solving for \( BC \)**:
   Now that we know \( AB = 8 \), we can substitute it back into the proportion:
   \[
   \frac{8}{8} = \frac{6}{x}
   \]
   This simplifies to:
   \[
   1 = \frac{6}{x}
   \]
   Solving for \( x \):
   \[
   x = 6
   \]

Thus, the length of \( BC \) is 6 cm.

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

### Related Questions:
1. How do you determine if two triangles are similar?
2. Can triangle similarity be used to solve any kind of triangle problem?
3. What is the Pythagorean theorem, and how does it apply to this problem?
4. How do you calculate the length of a chord in a circle?
5. What are the properties of a circle that help in solving geometry problems?

### Tip:
Always look for symmetry and similarity in geometry problems—they often simplify the process of solving for unknown lengths.

A, B, and C are three points on a circle, center O, with a radius of 8 cm. AE = 10 cm and OE = 6 cm. Given that triangle AOE is similar to triangle ABC, find BC.

Learn how to find the length of BC in a circle using triangle similarity and the Pythagorean theorem. This problem involves applying circle theorems and triangle similarity concepts to solve for unknown lengths.

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