To find the coordinates of the incenter of a triangle, we can use the formula:

$I(x, y) = \left( \frac{aA_x + bB_x + cC_x}{a + b + c}, \frac{aA_y + bB_y + cC_y}{a + b + c} \right)$

Where:

• $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ are the coordinates of the triangle's vertices.
• $a$, $b$, and $c$ are the lengths of the sides opposite to vertices $A$, $B$, and $C$, respectively.

Step 1: Find the lengths of the sides

We can use the distance formula to calculate the lengths of the sides: $\text{Distance between two points} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

1. Side $a$ (opposite vertex $A$): Between points $B(-2, 4)$ and $C(5, 5)$: $a = \sqrt{(-2 - 5)^2 + (4 - 5)^2} = \sqrt{(-7)^2 + (-1)^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}$

2. Side $b$ (opposite vertex $B$): Between points $A(4, -2)$ and $C(5, 5)$: $b = \sqrt{(4 - 5)^2 + (-2 - 5)^2} = \sqrt{(-1)^2 + (-7)^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2}$

3. Side $c$ (opposite vertex $C$): Between points $A(4, -2)$ and $B(-2, 4)$: $c = \sqrt{(4 - (-2))^2 + (-2 - 4)^2} = \sqrt{(6)^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}$

Step 2: Apply the formula for the incenter

We now substitute the values into the formula for the incenter:

$I(x, y) = \left( \frac{aA_x + bB_x + cC_x}{a + b + c}, \frac{aA_y + bB_y + cC_y}{a + b + c} \right)$

Substitute the known values:

$a = 5\sqrt{2}, \quad b = 5\sqrt{2}, \quad c = 6\sqrt{2}$ $A(4, -2), \quad B(-2, 4), \quad C(5, 5)$

$x = \frac{5\sqrt{2} \times 4 + 5\sqrt{2} \times (-2) + 6\sqrt{2} \times 5}{5\sqrt{2} + 5\sqrt{2} + 6\sqrt{2}} = \frac{20\sqrt{2} - 10\sqrt{2} + 30\sqrt{2}}{16\sqrt{2}} = \frac{40\sqrt{2}}{16\sqrt{2}} = 2.5$

$y = \frac{5\sqrt{2} \times (-2) + 5\sqrt{2} \times 4 + 6\sqrt{2} \times 5}{5\sqrt{2} + 5\sqrt{2} + 6\sqrt{2}} = \frac{-10\sqrt{2} + 20\sqrt{2} + 30\sqrt{2}}{16\sqrt{2}} = \frac{40\sqrt{2}}{16\sqrt{2}} = 2.5$

Thus, the coordinates of the incenter are (2.5, 2.5).

Do you need further details or have any questions about this process?

Here are some related questions you might find interesting:

1. How do you calculate the area of a triangle using its incenter?
2. What is the relationship between the incenter and the circumcenter of a triangle?
3. How do you determine the area of a triangle given its side lengths?
4. What is the importance of the incenter in geometric constructions?
5. How do you calculate the coordinates of the centroid of a triangle?

Tip: The incenter is always equidistant from the sides of the triangle, as it lies on the angle bisectors.