Using the formula for the radius of the incircle r = A/s, where A is the area and s is the semi-perimeter.

Using the distance formula, the radius can be calculated from the center found using the circumcircle method.

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

The lines are parallel if the discriminant of the quadratic equation is zero, which leads to the condition h^2 = ab.

The lines are perpendicular if 2h = a + b, which leads to h^2 = -ab.

For the lines to be coincident, the discriminant must be zero, i.e., b^2 - 4ac = 0.

The lines are perpendicular if the condition a + b = 0 holds true.

Equation of circle: (x - h)² + (y - k)² = r² => (x - 2)² + (y + 3)² = 5² = 25.

The slope m = (9 - 0) / (3 - 0) = 3. The equation is y = 3x.

The equation x^2 - y^2 = c represents a family of hyperbolas with varying values of c.

The equation x^2/a^2 + y^2/b^2 = 1 represents a family of ellipses with varying semi-major and semi-minor axes.

The equation y = ax^2 + bx + c represents a family of parabolas with varying coefficients a, b, and c.

The equation y = ax^3 + bx represents a family of cubic functions where a and b are constants.

The equation y = ax^3 + bx^2 + cx + d represents a family of cubic functions.

The equation y = e^(kx) represents a family of exponential curves with varying growth rates determined by k.

The equation y = kx^2 represents a family of parabolas that open upwards or downwards depending on the sign of k.

The standard form of the parabola is x^2 = 4py. Here, 4p = 8, so p = 2. The focus is at (0, p) = (0, 2).

The length of the latus rectum for the parabola y^2 = 4px is given by 4p. Here, p = 4, so the length is 16.

The discriminant indicates that the lines intersect at two distinct points.

Setting y = 0 in the equation gives 4x + 8 = 0, thus x = -2.

Setting y = 0 in the equation gives 4x + 20 = 0, thus x = -5.

Setting y = 0 in the equation gives 5x - 10 = 0, thus x = 2.

Set y = 0 in the equation: 2x + 6 = 0 => x = -3.

The angle between the lines can be found using the formula tan(θ) = |(m1 - m2) / (1 + m1*m2)|, where m1 and m2 are the slopes of the lines. The slopes can be found from the equation.

The slopes are m1 = 2 and m2 = -0.5. The angle θ is given by tan(θ) = |(m1 - m2) / (1 + m1*m2)| = |(2 + 0.5) / (1 - 1)|, which is undefined, indicating 90 degrees.

Area = 1/2 * base * height = 1/2 * 4 * 3 = 6.

For the lines to be parallel, the discriminant D must be equal to 0.

The condition for the lines to be parallel is given by h^2 = ab.

Centroid = ((x1+x2+x3)/3, (y1+y2+y3)/3) = (9/3, 6/3) = (3, 2).

Centroid = ((1+3+5)/3, (2+4+6)/3) = (3, 4).