If A(4, 3), B(0, 0) and C(2, 3) are the vertices of a `triangle ABC`, find the equation of the bisector of `angle A`.

Answer

To find the equation of the angle bisector of angle A in triangle ABC with vertices A(4, 3), B(0, 0), and C(2, 3), we can follow these steps:

### Step 1: Find the equations of lines AB and AC

1. **Equation of line AB**:
- Points A(4, 3) and B(0, 0).
- Using the two-point form of the equation of a line:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{x - x_1}{x_2 - x_1} \]
- Substituting A(4, 3) and B(0, 0):
\[ \frac{y - 0}{3 - 0} = \frac{x - 0}{4 - 0} \]
- This simplifies to:
\[ \frac{y}{3} = \frac{x}{4} \implies 4y = 3x \implies 3x - 4y = 0 \]
- Thus, the equation of line AB is \(3x - 4y = 0\).

2. **Equation of line AC**:
- Points A(4, 3) and C(2, 3).
- Since both points have the same y-coordinate, the line is horizontal:
\[ y = 3 \]
- This can be written as \(y - 3 = 0\).

### Step 2: Use the angle bisector formula

The angle bisector of two lines can be found using the formula:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]
where \(A_1, B_1, C_1\) are coefficients from line AB and \(A_2, B_2, C_2\) are coefficients from line AC.

- For line AB \(3x - 4y = 0\):
- \(A_1 = 3\), \(B_1 = -4\), \(C_1 = 0\)

- For line AC \(y - 3 = 0\):
- \(A_2 = 0\), \(B_2 = 1\), \(C_2 = -3\)

### Step 3: Substitute into the angle bisector formula

Substituting into the formula gives:
\[ \frac{3x - 4y}{\sqrt{3^2 + (-4)^2}} = \pm \frac{0 \cdot x + 1 \cdot y - 3}{\sqrt{0^2 + 1^2}} \]
Calculating the denominators:
- \(\sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5\)
- \(\sqrt{0^2 + 1^2} = 1\)

Thus, we have:
\[ \frac{3x - 4y}{5} = \pm (y - 3) \]

### Step 4: Solve for the two cases

1. **Case 1**:
\[ \frac{3x - 4y}{5} = y - 3 \]
Multiplying through by 5:
\[ 3x - 4y = 5y - 15 \implies 3x - 9y + 15 = 0 \implies 3x - 9y = -15 \implies x - 3y = -5 \]

2. **Case 2**:
\[ \frac{3x - 4y}{5} = -(y - 3) \]
Multiplying through by 5:
\[ 3x - 4y = -5y + 15 \implies 3x + y - 15 = 0 \implies 3x + y = 15 \]

### Final Result

Thus, the equations of the angle bisectors of angle A are:
1. \(x - 3y = -5\)
2. \(3x + y = 15\)