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Triangles from the Feuerbach Triangle

Zvonko Čerin ; PMF, Matematički odjel Sveučilišta u Zagrebu, Zagreb, Croatia

Sažetak

We prove four unusual theorems about the Feuerbach triangle A_dB_dC_d of the given triangle ABC whose vertices are points A_d, B_d, and C_d in which the excircles touch from outside the nine-point circle. These results concern the problem to determine for which triangles XYZ will the segments A_dX, B_dY, and C_dZ be sides of a triangle. We shall find five triangles XYZ (including the degenerate triangle at the point D in which the incircle touches from inside the nine-point circle) associated to a triangle ABC such that A_dX, B_dY, and C_dZ are never sides of a triangle. On the positive side, we discover three triangles XYZ such that the segments A_dX, B_dY, and C_dZ are always sides of a triangle. We give an algebraic method of proof for these results based on simple analytic geometry in the plane. We also show how one can discover these and other related results using the Geometer's Sketchpad.

Ključne riječi

triangle; incircle; excircles; nine-point circle; Feuerbach triangle; central points; Feuerbach point; Geometer's Sketchpad

Hrčak ID:

4002

URI

https://hrcak.srce.hr/4002

Datum izdavanja:

19.2.2002.

Podaci na drugim jezicima: hrvatski

Posjeta: 1.764 *