# Analytic Geometry: Triangles and Conic Sections

This page continues with **analytic geometry**, covering triangles defined by coordinates and introducing conic sections.

For triangles defined by coordinates (x₁, y₁), (x₂, y₂), and (x₃, y₃), the following are presented:

- Area formula using the "basket method"
- Area formula using matrix determinants
- Coordinates of the centroid: xc = (x₁ + x₂ + x₃)/3, yc = (y₁ + y₂ + y₃)/3

**Vocabulary**: The centroid of a triangle is the point where its three medians intersect, dividing each median in a 2:1 ratio.

The page then introduces the general form of conic sections:
Ax² + Bxy + Cy² + Dx + Ey + F = 0

When B = 0, the equation simplifies to:
Ax² + Cy² + Dx + Ey + F = 0

**Definition**: Conic sections are curves formed by the intersection of a plane and a double cone. They include circles, ellipses, parabolas, and hyperbolas.

The type of conic section can be determined based on the coefficients:

- If A = 0 or C = 0, it's a parabola
- If A = C ≠ 0, it's a circle
- If A ≠ C and same sign, it's an ellipse
- If A ≠ C and different signs, it's a hyperbola

**Example**: The equation x² + y² = 25 represents a circle with radius 5 centered at the origin.

**Highlight**: Understanding conic sections is crucial for many applications in physics, engineering, and advanced mathematics.