A Generalization of the Collinearity of the Stereographic Projection around the Lemoine Point

Vladislav NatchevAmerican College of Sofia (Bulgaria)

https://doi.org/10.53656/math2025-4-1-gcs

Abstract. Under stereographic projection, the projection point is collinear with the Lemoine points of the projection and the projected triangles or with the centers of their Apollonian circles (Natchev 2025). In the current paper, we generalize the discovered property of stereographic projection by proving that it applies to every point on the plane expressed in the barycentric form (δa2 : εb2 : ρc2), where δ, ε, ρ ∈ R. For a particular case, we investigate the points on the symmedians of the triangle and the tangents to the circumcircle at the vertices, where we derive collinearity of two more notable points of a triangle, namely the feet of the symmedians and the vertices of the tangential triangle. By setting the planimetric equivalent of the newly found facts, we connect the configurations they give rise to with Olympiad geometry.Keywords: stereographic projection, collinearity, Olympiad geometry

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