GEOGEBRA

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.