An Investigation ofAngle Bisectors in a Triangle

Background

In this section of the exploration you will be asked to discover the way the angle bisectors of a triangle relate to each other and to the sides of the triangle. As you work through the material, you may be asked to write down the answers to the questions in the investigation or to save the information and your sketch on a disk. Ask your teacher for specific directions.

Exploring the Properties of the Angle Bisector

The angle bisector of an angle ABC is the ray BE in the interior of angle ABC such that the measure of angle ABE is equal to the measure of angle CBE.

Click here once to play with the definition of an angle bisector. After you are done close the sketch without saving it.

Investigation

Let us recall that the distance between a line and a point not lying on the line is the length of the perpendicular segment drawn from the point to that line. We will compare the distances from a point on the angle bisector to each of the sides of angle ABC.

Click on this sketch

• Click the show button twice to display the distances from the point E to the sides of the angle bisector. What do you observe about the lengths of the segments?
• Move the point E along the angle bisector. What do you observe is happening to the two perpendicular distances?
• Close the sketch without saving it, unless otherwise requested.

Now we are going to investigate the properties of the angle bisectors of a triangle. In the picture below the intersection point of the angle bisectors of angle CAB and angle BCA is denoted by M. Click on the picture.

• Construct the angle bisector of angle ABC.
• Make an observation about the position of M with respect to the third angle bisector.
• Drag the points A, B and C. Does your observation still hold?
• Close the sketch without saving it, unless otherwise requested.

Now we are going to investigate how far is M from the sides of triangle ABC. Is M closer to some sides and further away from others? Do you have any idea what the answer might be? Remember that we are studying the lengths of the perpendicular segments from M to the sides and that M lies on the angle bisectors.

Let us start our investigation by clicking here to activate a sketch.

• The sketch represents a triangle with two angle bisectors intersecting at point M.
• Click the show button twice and then on the sketch.
• The Geometer's Sketchpad has constructed the perpendicular segments from M to the sides AB, BC and AC and it also has displayed their lengths.
• Move the vertices of triangle ABC to see what happens to the lengths. Do they stay the same?
• Construct a circle with center point M and radius one of the perpendicular segments.
• Close the sketch without saving it, unless otherwise requested.

The angle bisectors of a given triangle intersect at a point called the incenter of the triangle. The incenter is the center of the circle inscribed in the triangle.

Click here to activate a new Sketch. It displays a triangle and its inscribed circle.

After we have studied the properties of the angle bisectors in a triangle, the inscribed circle and the incenter, we will turn our attention to the construction of externally tangent circles.