In a `Delta`ABC, I is the incentre and AI cuts BC at D. If B=(-1,2) C(9,7), AB = 14, AC =21 then find D.

Answer

To find the coordinates of point D where the angle bisector AI intersects side BC in triangle ABC, we can follow these steps:

### Step 1: Identify the coordinates of points B and C.
Given:
- B = (-1, 2)
- C = (9, 7)

### Step 2: Calculate the lengths of sides AB and AC.
We are given:
- AB = 14
- AC = 21

### Step 3: Determine the ratio in which point D divides line segment BC.
According to the angle bisector theorem, the internal angle bisector divides the opposite side in the ratio of the lengths of the other two sides. Hence, we have:
- Ratio of AB to AC = 14:21 = 2:3

### Step 4: Use the section formula to find the coordinates of point D.
The section formula states that if a point D divides the line segment joining points B(x1, y1) and C(x2, y2) in the ratio m:n, then the coordinates of D are given by:
\[ D\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
Here, m = 2 and n = 3, and we have:
- \(x_1 = -1\), \(y_1 = 2\) (coordinates of B)
- \(x_2 = 9\), \(y_2 = 7\) (coordinates of C)

Substituting the values into the formula:
\[ D_x = \frac{2 \cdot 9 + 3 \cdot (-1)}{2 + 3} = \frac{18 - 3}{5} = \frac{15}{5} = 3 \]
\[ D_y = \frac{2 \cdot 7 + 3 \cdot 2}{2 + 3} = \frac{14 + 6}{5} = \frac{20}{5} = 4 \]

### Step 5: Conclude the coordinates of point D.
Thus, the coordinates of point D are:
\[ D = (3, 4) \]