A remarkable incidence theorem with applications to triangle geometry
The author gives three different proofs for the validity in elementary geometry of an apparently new configuration theorem, which reads as follows: If (a) the triplets \(A_ 1\), \(A_ 2\), \(A_ 3\); \(B_ 1\), \(B_ 2\), \(B_ 3\), \(B_ 1'\), \(B_ 2'\), \(B_ 3'\) are not collinear; (b) \(B_ i,B_ i'\in A_{i+1} \setminus \{A_{i+1}, _{i+2}\}\); (c) \(C_ i= A_ i B_ i'\cap B_{i+1} B_{i+2}\), \(C_ i'= A_ i B_ i\cap B_{i+1}' B_{i+2}'\); (d) \(S_ i= B_{i+2}\cap B_ i' B_{i+1}'\), \(S_ i'= B_ i' B_{i+2}'\cap B_ i B_{i+1}\); (e) the lines \(B_ 1 C_ 1\), \(B_ 2 C_ 2\), \(B_ 3 C_ 3\) are concurrent in \(P\); then, for all \(i=1,2,3\), the following hold: (1) \(P\in B_ i' C_ i'\); (2) \(P\in A_ i S_ i\); (3) \(P\in A_ i S_ i'\); and (4) the points \(B_ 1\), \(B_ 2\), \(B_ 3\), \(B_ 1'\), \(B_ 2'\), \(B_ 3'\) lie on a conic section. The author concludes by establishing a relationship between this configuration theorem and several special points of a triangle (the circumcenter, the Feuerbach, Gergonne and Lemoine points).
scientific article; zbMATH DE number 495250