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The image contains a geometry problem with three equations representing the sides of triangle \( \triangle ABC \), and two tasks:

**Equations of sides:**
- \( AB: x + y = 0 \)
- \( BC: x - 4y = 0 \)
- \( AC: 3x + 8y + 40 = 0 \)

**Task (a):**
Find the coordinates of points \( A \), \( B \), and \( C \).

**Task (b):**
Find the coordinates of the circumcenter of \( \triangle ABC \).

### Solution for (a) - Finding Coordinates of \( A \), \( B \), and \( C \)

1. **Find the intersection of lines AB and AC to get point \( A \):**
   Solve the system of equations \( x + y = 0 \) and \( 3x + 8y + 40 = 0 \).
   
2. **Find the intersection of lines AB and BC to get point \( B \):**
   Solve the system of equations \( x + y = 0 \) and \( x - 4y = 0 \).

3. **Find the intersection of lines BC and AC to get point \( C \):**
   Solve the system of equations \( x - 4y = 0 \) and \( 3x + 8y + 40 = 0 \).

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Would you like to proceed with the detailed steps for finding these coordinates?

Here are five related questions:
1. What is the general method to find the circumcenter of a triangle?
2. How do we interpret the circumcenter geometrically?
3. What is the significance of the perpendicular bisectors in finding the circumcenter?
4. How do we solve systems of linear equations involving two variables?
5. Can we use alternative methods like matrices to solve these systems?

**Tip:** The circumcenter is equidistant from all three vertices of the triangle, as it is the point where the perpendicular bisectors of the triangle meet.

The equations of AB, BC, and AC are x + y = 0, x - 4y = 0, and 3x + 8y + 40 = 0 respectively. (a) Find the coordinates of A, B, and C. (b) Find the coordinates of the circumcenter of triangle ABC.

Solve a geometry problem involving finding the coordinates of the vertices of a triangle and its circumcenter. Learn how to solve systems of linear equations and apply the properties of the circumcenter in triangle ABC.

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