The incenter of a triangle is the point where the angle bisectors of the triangle intersect. In a right-angled triangle, the incenter can be found using a simple method based on the lengths of the sides.
To find the incenter (I) of a right-angled triangle, follow these steps:
- Identify the lengths of the sides of the triangle. Let’s denote the lengths of the sides opposite to angles A, B, and C (where C is the right angle) as a, b, and c respectively. Here, c is the hypotenuse.
- Use the formula for the coordinates of the incenter:
I_x = (a * x_A + b * x_B + c * x_C) / (a + b + c)
I_y = (a * y_A + b * y_B + c * y_C) / (a + b + c)
- Replace x_A, y_A, x_B, y_B, x_C, and y_C with the coordinates of the vertices A, B, and C of the triangle.
For example, if you have a right-angled triangle at points A(0, 0), B(4, 0), and C(0, 3), the lengths of the sides would be a = 3, b = 4, and c = 5.
Plugging the values into the incenter formula:
I_x = (3 * 0 + 4 * 4 + 5 * 0) / (3 + 4 + 5) = 16 / 12 = 4/3
I_y = (3 * 0 + 4 * 0 + 5 * 3) / (3 + 4 + 5) = 15 / 12 = 5/4
Therefore, the incenter I is located at the coordinates (4/3, 5/4). This point is equidistant from all three sides of the triangle, making it the center of the inscribed circle.