Megan Barnett

Raju Patel

Corrie Collier

For this assignment, we were to construct the Feurerbach point. The Feurbach point is the point of tangency of the nine-point circle and the incircle. In constructing the Feurerbach point, we…

Link to GSP Link to more info

1. Constructed a triangle
2. Constructed the incenter of the triangle
3. Inscribed a circle inside the triangle using the incenter as the center

 



4. Constructed the altitudes of the triangle

 

 



 

5. Construct the circumcenter of the triangle. From the orthocenter (point F) draw line to circumcenter and then find and construct the midpoint (point I)

 



 

5. Construct the midpoint of one side of the triangle. From that midpoint to point I, construct a circle using point I as the center. Use segment IM as the radius.

6. Now, we have constructed Feurerbach’s point (F’)…tangency of the nine-point circle and the incircle.

In exploring our triangle and Feurerbach point, we came to several conclusions...

1. The center of the big circle is the same distance to the circumcenter and to the

orthocenter of the original triangle. Therefore, the distance is the length of the

midpoint of the orthocenter and circumcenter. Our conjecture is that the distance

will remain the same no matter what.

2. When you construct an equilateral triangle with the given triangle, you will notice

that both of the circles lie directly on top of each other and all the points are the exact same. Consequently, the Feurerbach’s point could be infinite points on the circles.

3. Finally, we explored an isosceles triangle with our construction. We established

Here, we will prove our conjecture that when you construct an equilateral triangle with the given construction, Feurerbach’s point could have infinite solutions…

As you can see, in constructing an equilateral triangle the two circles end up lying on top of one another making them the exact same. Therefore, there is no exact point where the two circles come together and Fuererbach’s point can have infinite solutions.

The idea of Feurerbach’s point did not just appear out of nowhere. In the early 1800’s, a man by the name Karl Feurerbach introduced the Feuerbach point of a triangle. Karl Feurbach was born to a distinguished German family on May 30, 1800. He later went on to become a brilliant student and was appointed to a professorship at the Gymnasium at Erlangen. During his teaching career, Feurerbach published two books including one that introduced his theorem of the Feurerbach’s point…Eigenschaften einiger merkwurdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren: analytisch-trigonometrische Abhandlung (Properties of some special points in the plane of a triangle, and various lines and figures determined by these points: an analytic-trigonometric treatment). However, his teaching career did not last long. After six years of his profession, Feurerbach had to retire after threatening his students with a sword but most importantly, he had to retire because he was unable to cope any longer given the serious illness he had. Six years later, Karl Feurerbach died at the age of 34.

An extension to this investigation that might be amusing to explore would be to figure out how to construct the excircles. The excircles are the three circles that are tangent to all three sides (or their extensions) of the given triangle.