Geometry » Special lines

Contents

1. Angle bisector
2. Perpendicular bisector
3. Median
4. Altitude
5. Mid-parallel

1. Angle bisector

The angle bisector divides an angle into two equal parts.
angle with bisector drawn

To construct an angle bisector you need a pair of compasses and a ruler.

Click here to show how this is done.

The three angle bisectors in a triangle will intersect in one point. This is shown in the picture on the left.
Triangle with the three angle bisectors drawn                 Triangle with the three angle bisectors drawn including the incircle

Incircle

If you would draw a circle with centre S and radius the shortest distance from S to one of the sides, you will get the incircle of the triangle.
See this in GeoGebra!

2. Perpendicular bisector

The perpendicular bisector between two points is a line segment that intersects the line segment between those two points in the middle at an angle of 90 °.
Below you can see the perpendicular bisector of line segment AB.
Every point on the perpendicular bisector is equidistant to A and B.
Perpendicular bisector drawn between point A and B

To construct a perpendicular bisector you need a pair of compasses and a ruler.

Click here to show how this is done.

The three perpendicular bisectors of a triangle will intersect in one point. This is shown in the picture on the left.
In case of an obtuse-angled triangle this intersection will be outside the triangle!

Triangle with all three perpendicular bisectors drawn  Triangle with all three perpendicular bisectors drawn including the circumcircle

Circumcircle

You can use the intersection of the perpendicular bisectors to draw the circumcircle.
See this in GeoGebra! You can also see what happens with an obtuse-angled triangle.

3. Median

The median of a triangle is a line through one of the vertices of the triangle to the middle of the opposite side of this vertex.
The area on both sides of the median is equal. For that reason the weight of the triangle on both sides is also equal, therefore you can balance the triangle on that line. Small technicality: when a triangle has weight you actually have a prism.
A triangle with one median drawn

Centroid

The three medians in a triangle will intersect in one point.
This point is called the centroid.
The centroid is the centre of gravity of the triangle.
For that reason you can balance the triangle on a sharp tip by putting the triangle with its centroid exactly on the sharp tip.
Triangle with all three medians drawn
The medians intersect each other with a ratio of 1 : 2.
In the figure above, the red part is always half as long as the green part of the median starting in point B.
See this in GeoGebra!

4. Altitude

An altitude is the shortest distance between a certain point and a line. The shortest distance is always perpendicular to the line. Sometimes the altitude is called the perpendicular.
Below you can see two altitudes.
Obtuse-angled triangle with two altitudes drawn, one inside the triangle and one outside the triangle

Orthocentre

The three altitudes in a triangle will intersect in one point.
This point is called the orthocentre.
In case of an obtuse-angled triangle, the orthocentre will be outside of the triangle.
Acute-angled triangle with the three altitudes drawn
See this in GeoGebra!
You can also see what happens with an obtuse-angled triangle.

5. Mid-parallel

A line that connects the middle of one side with the middle of another side in a triangle is called a mid-parallel. Every mid-parallel is parallel to one of the sides in the triangle and is half the size of the side it runs parallel with.
Triangle with one mid-parallel drawn
In the figure above DE is parallel to BC and DE is half as long as BC.
See this in GeoGebra!


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