# Circumcenter

- Author:
- Jorge Cássio

## 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?