Circumcenter
Circumcenter
The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of a triangle intersect.
Constructing the Triangle Circumcenter
- Enable the tool POLYGON (Window 5) and click on three different places to form a triangle. In order to close the triangle click on the first point again. Naturally, the points cannot be aligned. A triangle with vertices at points A, B and C will be appear.
- Enable the tool PERPENDICULAR BISECTOR (Window 4). By clicking on side c (which joins point A to B), a line d will appear; By clicking on side b, a line will appear.
- Select the tool INTERSECT (Window 2) and click on the lines d and e. Point D will appear.
- Similarly to the previous activity, we ask: will the other line segment bisector also pass through point D? In order to prove this argument, click again on side a of the triangle, with the tool PERPENDICULAR BISECTOR (Window 4) enabled. A line f will appear. Check if this last line also passed through point D.
Constructing the circumscribed circle
In the previous app
- Enable the tool DISPLAY / HIDE OBJECT (Window 12), click on the lines d, e and f and then press ESC.
- Let us change the name of point D to Circumcenter. In order to do this, right click the mouse on point D and check the option RENAME. In the new open window, type Circumcenter and click OK.
- Enable the tool CIRCLE CENTER THROUGH POINT (Window 6), click on the Circumcenter point and, then on one of the vertices of the triangle. A circle g will appear. What do you notice?
Analysis 1
Why does the circle passes through the three vertices of the triangle?
Analysis 2
When will the triangle have an internal circumcenter? When will the triangle have an external circumcenter? When will the circumcenter be on one side of the triangle?
Property
The perpendicular bisectors of the three sides of a triangle are concurrent (intersect) in a point that is equidistant (the same distance) from the vertices of the triangle.
Proof
Analysis
Do you understand the theorem's proof?