## GeoGebra Tutorial: Using the Perpendicular Line and the Point on Object Tools

In this tutorial, we use GeoGebra to explore the minimum sum of the distances of a point on the interior of a triangle to its sides. We learn how to use the Point on Object tool and the Perpendicular Line tool. We also learn how to compute using the Input bar. The final output of this tutorial is shown in the following applet.
This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com
Steps in Creating the Applet

 No Instruction 1 Using the Polygon tool, construct triangle ABC. 2 Select the Point on Object tool, and then click the interior of the triangle to create point D. 3 Move point D and observe what happens. Can you move point D outside the triangle? 4 Next, we create a line perpendicular to AB passing through D.  To do this, select the Perpendicular Line tool, select segment AB, and then select point D. 5 Create two more lines passing through D, one perpendicular to AC and the other perpendicular to BC. 6 Using the Intersect Two Objects tool, intersect the lines and the perpendicular segments. Your drawing should look like the figure below. 7 Hide the three lines by right clicking each line and clicking them and selecting Show Objects. 8 Using the Segment between Two Points tool, construct segments DE, DG, and DF. 9 To find the total distance t of points D from E, F, and G, type t = g + h + i, and then press the ENTER key. 10 Move the points to explore and observe the value of t.

Application

A building is going to be constructed with a triangle bounded by the roads connecting Buildings A, B, and C. A connecting pathway is to be constructed from the three roads to the building. Find the location of the building that will minimize the cost of pathway construction.

## Circumcircle and Circumcenter

Click the check boxes and move the vertices of the triangle to explore. What do you observe?

In any triangle, a circle called circumcircle maybe drawn that passes through its vertices. The center of the circumcircle is called the circumcenter. The perpendicular bisector of the three sides of any triangle passes through its circumcenter.

## Napoleon’s Theorem

In the applet below, ABC is a triangle whose sides are also sides of equilateral triangles that contain the green line segments. Points P, Q, and R are centroids of the equilateral triangles.

1) What do you observe about the figure?
2) Move points A, B, and C. Are your observations still the same?

This is a Java Applet created using GeoGebra from www.geogebra.org – it looks like you don’t have Java installed, please go to www.java.com

The Napoleon theorem states that if equilateral triangles are constructed on the sides of any triangle, either all outward, or all inward, the centers of those triangles themselves form an equilateral triangle.