Angle Bisectors and Incircle

🔮 Angle Bisector Definition:

An angle bisector is a line that splits an angle exactly in half, creating two equal angles.

• The angle bisector always starts at the angle’s vertex (corner) and cuts the angle into two equal parts.
• For example, if you draw an angle bisector of a 60° angle, you create two equal angles of 30° each.

There are two common ways to draw an angle bisector:

🔮 Drawing an Angle Bisector With a Protractor

1. Measure the angle with your protractor. For example, suppose it’s 40°.

2. Divide 40° by 2, which is 20°. Use your protractor to mark 20° from one side of the angle.

3. Finally, draw a line from the vertex through this mark. And that’s your angle bisector!

🔮 Drawing an Angle Bisector With a Compass

• Place the compass point at the angle’s vertex and draw an arc that crosses both sides of the angle.

• Without changing the compass width, place the compass on each intersection point and draw two more arcs.

• Draw a straight line from the corner through where the arcs meet. And that’s your angle bisector!

Exam tip: Using a protractor is quick and easy because you just measure the angle. But if you don’t have a protractor, a compass is a classic method that works just as well:)

🔮 A Special Property of Angle Bisectors

A special property of the angle bisector is that every point on the bisector is the same distance from both sides of the angle.

If you pick any point on the bisector line, and draw the perpendicular distances to each side so they meet at a right angle, both those distances will be the same length.

You need to know this property because it often helps solve geometry problems and is used in many constructions and proofs.

The incenter of a triangle is the point where all three angle bisectors meet.

To find the incenter, just draw the angle bisector of each corner of the triangle. The spot where all three bisectors cross is the incenter.

The incenter is always equally distant from all three sides of the triangle.

🔮 What is the Incircle of a Triangle?

The incircle is the largest circle that fits perfectly inside a triangle and just touches all three sides.

🔮 Steps to Draw the Incircle of a Triangle:

1. Find the incenter by drawing the angle bisectors of all three corners.

2. From the incenter, draw a straight line to one side of the triangle so that it meets the side at a right angle. Measure this length; this is the inradius.

3. Using a compass, draw a circle with the incenter as the center and the inradius as the radius. And there you have the triangle’s incircle.