Constructing
Medians and constructing the Centroid of a triangle
(A median
is a line segment connecting any vertex of a triangle to the midpoint of the opposite
side)
1.
Click
on File and select New
Window.
2.
Draw a
triangle using the Polygon tool as above, and using the Midpoint or
Centre tool and Segment between Two Points tool, construct the
medians of each side of the triangle.
3.
Construct
the intersection of the medians by selecting the Intersect Two Objects tool.
4.
Drag
any of the vertices of the
triangle and note that the 3 medians remain concurrent, at the CENTROID.
5.
Click
on File, Save as
and choose the location you wish to save your GeoGebra file to. Enter a file name
that describes the file content and click on Save.
6.
To
start a new file, click on File, New.
Constructing
Mediators and constructing the circumcentre and circumcircle of a triangle
(A
mediator is a perpendicular bisector of a line segment)
1.
Click
on File, New Window, and draw a triangle using the Polygon
tool .
2.
Select
the Midpoint or Centre tool and selecting each side of the triangle in
turn, construct the midpoints of each side.
3.
Using
the Perpendicular Bisector tool, select each side to construct
perpendicular bisectors (mediators) of each side.
4.
Select
the Intersect Two Objects tool, and then 2 of the mediators to construct
the circumcentre.
5. The equations of the 3 mediators are shown
in the Algebra View.
6. Hide
the mediators by right clicking on each one and clicking on Show
object. Drag the vertices to see the circumcentre change position.
7. Click
on the Circle with
Centre through Point tool, then the circumcentre (point of intersection of
the mediators) and one of the vertices of the triangle and construct the
circumcircle, which passes through the 3 vertices.
8.
Drag
the vertices of the
triangle to confirm the construction.
9.
Click
on File, Save as
and choose the location you wish to save your GeoGebra file to. Enter a filename
that describes the file content and click on Save.
10.
To
start a new file, click on File, New.
Constructing
the bisectors of the angles and constructing the incentre and incircle of a
triangle.
As in the last 2 examples, construct a triangle ABC in a new
window.
1. Select the Angle Bisector tool.
Select the points B, A and C, in that order, to construct the angular bisector
of <BAC. Repeat for the other two angles in the triangle.
2.
Select
the Intersect Two Objects tool and 2 of the angle bisectors to construct
the incentre.
3.
Hide
the angle bisector lines as in the previous example on the circumcircle.
4.
Each
side of the triangle will be a tangent to the incircle and should remain as
such if we were to drag the vertices. It is important therefore not to just use
the Circle with Centre through Point tool, selecting the incentre and
moving outwards until we touch the circle to draw the incircle.
5.
The
circle constructed in this way will not remain an incircle as we drag the
vertices. Try it to see.
To construct the incircle
1.
Selecting
the Perpendicular Line tool, draw a perpendicular line from the incentre
D, to line AB or any of the 3 sides of
the triangle.
2.
With
the Intersect Two Objects tool selected construct the intersection E of
side AB and this perpendicular line.
3.
Hide
the perpendicular line. Select the Circle with Centre through Point, and
with D as centre and E as the point on the circle, construct the incircle.
4.
Drag the vertices to confirm the
construction.
5.
Click
on File, Save as
and choose the location you wish to save your GeoGebra file to. Enter a filename
that describes the file content and click on Save.
6.
To
start a new file, click on File, New.
