{"id":165906,"date":"2025-09-25T15:08:50","date_gmt":"2025-09-25T12:08:50","guid":{"rendered":"https:\/\/azbuki.bg\/?p=165906"},"modified":"2025-10-06T09:50:20","modified_gmt":"2025-10-06T06:50:20","slug":"a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point","status":"publish","type":"post","link":"https:\/\/philosophy.azbuki.bg\/en\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/","title":{"rendered":"A Generalization of the Collinearity of the Stereographic Projection around the Lemoine Point"},"content":{"rendered":"<p><strong>Vladislav Natchev<\/strong><br \/>\n<em>American College of Sofia (Bulgaria)<\/em><\/p>\n<p><a href=\"https:\/\/doi.org\/10.53656\/math2025-4-1-gcs\" target=\"_blank\" rel=\"noopener\">https:\/\/doi.org\/10.53656\/math2025-4-1-gcs<\/a><\/p>\n<p><strong>Abstract.<\/strong> 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 (\u03b4a2 : \u03b5b2 : \u03c1c2), where \u03b4, \u03b5, \u03c1 \u2208 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.<br \/>\n<em>Keywords:<\/em> stereographic projection, collinearity, Olympiad geometry<\/p>\n<a href=\"https:\/\/philosophy.azbuki.bg\/en\/member-login\/\">Log in to read the full text<\/a>","protected":false},"excerpt":{"rendered":"<p>Vladislav Natchev American 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 [&hellip;]<\/p>","protected":false},"author":124332423427287,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jnews-multi-image_gallery":[],"jnews_single_post":[],"jnews_primary_category":[]},"categories":[1],"tags":[13465,10666,10662],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>A Generalization of the Collinearity of the Stereographic Projection around the Lemoine Point - \u0410\u0437-\u0431\u0443\u043a\u0438<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/bel.azbuki.bg\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A Generalization of the Collinearity of the Stereographic Projection around the Lemoine Point - \u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"og:description\" content=\"Vladislav Natchev American 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 [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/bel.azbuki.bg\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/\" \/>\n<meta property=\"og:site_name\" content=\"\u0410\u0437-\u0431\u0443\u043a\u0438\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/Azbuki55\/\" \/>\n<meta property=\"article:published_time\" content=\"2025-09-25T12:08:50+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-10-06T06:50:20+00:00\" \/>\n<meta name=\"author\" content=\"\u201e\u0410\u0437-\u0431\u0443\u043a\u0438\u201c\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"\u201e\u0410\u0437-\u0431\u0443\u043a\u0438\u201c\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/bel.azbuki.bg\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/bel.azbuki.bg\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/\"},\"author\":{\"name\":\"\u201e\u0410\u0437-\u0431\u0443\u043a\u0438\u201c\",\"@id\":\"https:\/\/azbuki.bg\/#\/schema\/person\/de220d282eaa494f914ce0fd838645dd\"},\"headline\":\"A Generalization of the Collinearity of the Stereographic Projection around the Lemoine Point\",\"datePublished\":\"2025-09-25T12:08:50+00:00\",\"dateModified\":\"2025-10-06T06:50:20+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/bel.azbuki.bg\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/\"},\"wordCount\":191,\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/#organization\"},\"keywords\":[\"collinearity\",\"Olympiad geometry\",\"stereographic projection\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/bel.azbuki.bg\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/\",\"url\":\"https:\/\/bel.azbuki.bg\/uncategorized\/a-generalization-of-the-collinearity-of-the-stereographic-projection-around-the-lemoine-point\/\",\"name\":\"A Generalization of the Collinearity of the Stereographic Projection around the Lemoine Point - 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