Altitudes and Orthocenters.

By Raynold Gilles.

Feurebach.

Above is a Picture of Feurebach who only live to be thirty four years of age. Feurebach came from a family of intellectual where his siblings were not afraid to pursue doctoral studies.

After attempting suicide, Feurebach lived a very isolated life where he discovered impressive properties of the nine point circle. The properties are usually referred to as the nine point circle properties.

Feurebach's conjecture (theorem) holds that the nine-point circle is tangent to each of the three excircles and the incircle. Our goal will be to proof this theorem. Feurebach's theorem is important in that it is linked to the generalization of the euler line.

Furthermore , Feurebach's conjecture helps geometers understand the relationship between the nine point circle and the nine point conic. Due to the nature of the proof and its complexity; I share two links that provides two different approaches to the proof.

__A GSP Construction of Feurebach Triangle.__

Feel free to click on the two links below to access the two proofs. Please do not hesitate to contact me or the authors should you have any questions on the proofs.

Click on the links below for a proof.

Click on the link below to manipulate the Script tool that allows you to consctruct a circle tangent to a given circle. Enjoy.

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