Definition

Internal angle bisector of a triangle

Geometry • Triangle • Angle Bisector

Statement

The internal bisector of a vertex angle divides it into two equal angles; the three internal bisectors of a triangle are concurrent at the incentre

I

, the point equidistant from the three sides, and the bisectors from

B

and

C

meet at

∠ B I C = 9 0^{\circ} + \frac{1}{2} ∠ A

How to use it

1Mark a bisected angle as two copies of one half-angle, and carry that half into whichever sub-triangle it sits in.
2In the triangle cut off by two bisectors, close the angles with the $18 0^{\circ}$ sum; $9 0^{\circ} + \frac{1}{2} ∠ A$ is the same answer as a one-line check.
3Keep the incentre apart from the circumcentre: the incentre is equidistant from the SIDES, the circumcentre from the VERTICES.

Worked example

In triangle

A B C

∠ A = 7 0^{\circ}

, and the bisectors of

∠ B

and

∠ C

meet at

I

. Find

∠ B I C

Step-by-step

1
$∠ B + ∠ C = 18 0^{\circ} - 7 0^{\circ} = 11 0^{\circ}$ , so the two half-angles sum to $5 5^{\circ}$
Why: Each bisector contributes half of its vertex angle to triangle $B I C$ .
2
$∠ B I C = 18 0^{\circ} - 5 5^{\circ} = 12 5^{\circ} = 9 0^{\circ} + \frac{7 0 ^{\circ}}{2}$

Answer

12 5^{\circ}

Watch out

An internal bisector is not in general the median or the altitude from the same vertex; the three coincide only at the apex of an isosceles triangle. The external bisector is a different line, perpendicular to the internal one.

Builds on

Angle sum of a triangle

Also known as

angle bisector, internal bisector halves the angle, incentre, bisectors of two angles of a triangle, 90 plus half A, BIC angle

Appears in past papers

Statement

Worked example

Appears in past papers

2020 (II)

2019 (II)

2017 (I)