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The image contains a problem focused on demonstrating that a set of given points are concyclic, meaning they all lie on a common circle.

### Here's a breakdown of the approach:

1. **General Circle Equation:**
   The equation of a circle is generally given by:
   \[
   x^2 + y^2 + 2gx + 2fy + c = 0
   \]
   Where \( g \), \( f \), and \( c \) are constants to be determined.

2. **Given Points:**
   The points provided are \( A(1, 2) \), \( B(3, -4) \), \( C(5, -8) \), and \( D(19, 8) \).

3. **Substitution into the Circle Equation:**
   To prove the points lie on the same circle, we substitute each point into the circle equation. By substituting the coordinates, a system of linear equations in terms of \( g \), \( f \), and \( c \) is formed.

4. **First Substitution (Point \( A(1, 2) \)):**
   Substitute \( (1, 2) \) into the general equation:
   \[
   1^2 + 2^2 + 2g(1) + 2f(2) + c = 0
   \]
   Simplifying this gives:
   \[
   1 + 4 + 2g + 4f + c = 0
   \]
   \[
   2g + 4f + c + 5 = 0 \quad \text{(Equation 1)}
   \]

5. **Second Substitution (Point \( B(3, -4) \)):**
   Substitute \( (3, -4) \) into the general equation:
   \[
   3^2 + (-4)^2 + 2g(3) + 2f(-4) + c = 0
   \]
   Simplifying this gives:
   \[
   9 + 16 + 6g - 8f + c = 0
   \]
   \[
   6g - 8f + c + 25 = 0 \quad \text{(Equation 2)}
   \]

6. **Third Substitution (Point \( C(5, -8) \)):**
   Substitute \( (5, -8) \) into the general equation:
   \[
   5^2 + (-8)^2 + 2g(5) + 2f(-8) + c = 0
   \]
   Simplifying this gives:
   \[
   25 + 64 + 10g - 16f + c = 0
   \]
   \[
   10g - 16f + c + 89 = 0 \quad \text{(Equation 3)}
   \]

### Next Steps:
You would typically solve this system of linear equations (Equations 1, 2, and 3) for \( g \), \( f \), and \( c \). After solving these, if the constants satisfy the equation when the fourth point \( D(19, 8) \) is substituted, all points are concyclic.

Would you like help solving the system of equations?

Show that the points (1,2), (3,-4), (5,-8), and (19,8) are concyclic.

Learn how to prove that the points (1,2), (3,-4), (5,-8), and (19,8) are concyclic. This problem involves the use of the general equation of a circle and linear algebra to determine if these points lie on the same circle. Suitable for Grades 10-12.

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