Triangle Bisector Calculator: How to Calculate and Why it Matters

A bisector is a ray emanating from the top of an angle and dividing this angle into two equal angles. You can also define the bisector as the geometrical location of the points inside the corner equidistant from the sides of this angle.

Triangle — is a geometric figure that has three points that do not lie on the same line and three segments that pair these points in pairs. The points of a triangle are usually called its vertices, and the segments are called its sides.

Intro

The bisectors of a triangle are lines that divide the interior angles of the triangle into two equal parts. These lines meet at a single point, known as the incenter, which is the center of the triangle's inscribed circle. The triangle bisector calculator allows you to quickly and easily calculate the bisectors of any triangle.

How to Calculate Triangle Bisectors

To calculate the bisectors of a triangle, follow these steps:

Through sides

• Measure the three sides of the triangle.
• Use the sides to calculate the bisector for each side using the following formula: l_a = \dfrac{\sqrt{cb(a+b+c)(c+b-a)}}{c+b} \\ l_b = \dfrac{\sqrt{ac(a+b+c)(a+c-b)}}{a+c} \\ l_c = \dfrac{\sqrt{ab(a+b+c)(a+b-c)}}{a+b} Where:
  • l_a, l_b, l_c - triangle bisectors
  • a, b, c - triangle sides

Through angle and side

• Measure the three angles of the triangle using a protractor and the sides.
• Calculate the half-angle for each angle by dividing it by 2.
• Use the half-angles to calculate the bisector for each angle using the following formula: l_a = \dfrac{a*sin(\frac{\gamma}{2})}{sin(90^0+\frac{\gamma}{2})} \\ l_b = \dfrac{b*sin(\frac{\beta}{2})}{sin(90^0+\frac{\beta}{2})} \\ l_c = \dfrac{c*sin(\frac{\alpha}{2})}{sin(90^0+\frac{\alpha}{2})}Where:
  • l_a, l_b, l_c - triangle bisectors
  • a, b, c - opposite sides to angles
  • \alpha, \beta, \gamma - triangle angles

Right triangle, through sides

• Measure the two sidess of the triangle.
• Use the sides to calculate the bisector using the following formula: l = \sqrt{2}\dfrac{ac}{a+c}Where:
  • l - triangle bisector
  • a, c - right triangle sides

Isosceles triangle, through angle and side

• Measure the one angle of the triangle and the opposite side to that angle.
• Use the angle and the side values to calculate the bisector using the following formula: l = m = h = asin(\alpha)Where:
  • l = m = h - in isosceles triangle bisector median and height are the same
  • a - isosceles triangle side
  • \alpha - isosceles triangle angle

Why Triangle Bisectors Matter

The bisectors of a triangle are important for a number of reasons. For example:

• They can be used to find the incenter of the triangle, which is the center of the circle that is tangent to all three sides of the triangle.
• They can be used to construct the medians of the triangle, which are lines that connect each vertex of the triangle to the midpoint of the opposite side.
• They are used in a variety of geometric proofs and constructions.

If you need to calculate the bisectors of a triangle, the triangle bisector calculator can help make the process quick and easy.