How the Angle Bisector Calculator Works

The Angle Bisector Calculator determines the coordinates where an angle bisector of a triangle intersects the opposite side. It uses the geometric properties of triangles and the Angle Bisector Theorem to calculate the results.

Inputs:

Coordinates of the Triangle Vertices: Enter the (x, y) coordinates of the three vertices of the triangle: A, B, and C.

How to calculate the angle bisector point:

The calculator uses the following steps to determine the point of intersection:

Calculate the side lengths: The lengths of sides AB, BC, and AC are computed using the distance formula:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Apply the Angle Bisector Theorem: The theorem states that the angle bisector divides the opposite side in the ratio of the adjacent sides:
Intersection Point (P) = ((x₂ × BC + x₃ × AB) / (AB + BC), (y₂ × BC + y₃ × AB) / (AB + BC))

The result provides the coordinates of the point (P) where the angle bisector intersects the opposite side.

Why Use the Angle Bisector Calculator?

Our Angle Bisector Calculator offers several benefits:

Accuracy: Uses precise calculations based on geometric principles.
Ease of Use: Simple interface requiring only vertex coordinates.
Detailed Steps: Displays intermediate calculations, including side lengths and ratios.

Examples of Angle Bisector Calculations with the Angle Bisector Calculator

Here are examples demonstrating the calculator's functionality:

Example 1

For a triangle with vertices A(0, 0), B(4, 0), and C(2, 3):

1. Calculate the side lengths:
   AB = √((4 - 0)² + (0 - 0)²) = 4.0
   BC = √((2 - 4)² + (3 - 0)²) = 3.6056
   AC = √((2 - 0)² + (3 - 0)²) = 3.6056

2. Apply the Angle Bisector Theorem:
   Intersection Point (P) = ((4 × 3.6056 + 2 × 4.0) / (4.0 + 3.6056), (0 × 3.6056 + 3 × 4.0) / (4.0 + 3.6056))
   Intersection Point (P) = (2.3333, 1.6667)

3. Result:
   The angle bisector intersects side BC at P(2.3333, 1.6667).