Fuerbach's Theorem.

Fuerbach's Theorem:

Nine special points of an arbitrary triangle ABC, in particular, the feet of the altitudes, the intersections of the medians with the opposite sides, and the midpoints of the segments from the vertices to the orthocenter, lie on a common circle, the so-called nine-point circle of ABC. Another result presently of interest to us is the concurrence of the three angle bisectors of ABC at a point, the incenter, that is the center of a circle, the incircle, tangent internally to each of the sides of ABC. Finally, a related result is that if the sides of ABC are extended, three additional circles, called excircles can be constructed, each tangent externally to the sides of ABC.

A German high-school teacher, K. Feuerbach, had the good fortune of discovering an amazing fact. For any triangle ABC, the nine-point circle is tangent to the incircle and to the three excircles.

 

To draw this diagram, first start JGEX. If the program is already running, use to start a new session.

1. Draw a Triangle . Use the action to draw an triangle( triangle ABC). See Action Triangle.

 

2. Three Midpoint . Use the action to draw the three midpoints of the three sides( point D, E, F). See Action Midpoint. (Note in JGEX, by default, the points in RED are free points and the points in GRAY are fixed points.)

 

3. The Foot . TUse the action to create a foot. Select the point A and drag to line BC, so that the foot G is generated. See action Foot.

 

4. Nine-Point Circle. Use the action to create the nine point circle. This circle is an circle which passes three points: D, E, F. With the right click menu, we can change the color of the circle to Red.

 

5. Incircle. (Please use the menu "Construct --> Point -->Incircle"). Use the action Incenter to create the incenter of triangle ABC. The incenter is named as H. From point H draw an foot line to BC. Draw a circle which takes H as the center and HI as the radius.

 

6. Exircles.

First draw three line AH, BH, CH, then use action Perpendicular Line (Please use the menu Construct -> Line -> Perpendicular Line) to draw three perpendicular line(The line in Gray) which is perpendicular to AH, BH, CH respectively. Change the color of the three lines to Gray.

Use the action to generate three intersected points. Then from these three points draw three foots to three sides respectively (KQ, LN, MP). Then we can draw the three excircle (in Green).

 

7. Decoration. We have finished the construction, however, we are still not satisfied with the diagram. We need to decorate the diagram according to our will.

  • Hiding the unecessary elements. Use the action Hide Object (Use the menu Action -> Hide Object) to hide unecessary elements. For example, we don't want the text of some points to be shown, so we hide all the text of point. Also the line AH, BH, CH are hidden as well.
  • Changing the color of some elements. The default color for the line and circle is Blue, we can use the Right-Click Menu to change the color of these elements.
  • Adding text. Use the action to add some text to the diagram. These text can be used to explain the theorem.
  • Freezing Points. In this diagram, the points A, B, and C are all free points and can be moved freely. If we want that point A to be the only point which can be moved freely, we can use Right- Click Menu to freeze points B and C. See Action Freeze.

The final diagram:

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