{"id":149027,"date":"2025-04-04T11:49:28","date_gmt":"2025-04-04T08:49:28","guid":{"rendered":"https:\/\/azbuki.bg\/?p=149027"},"modified":"2025-09-17T14:28:09","modified_gmt":"2025-09-17T11:28:09","slug":"apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point","status":"publish","type":"post","link":"https:\/\/foreignlanguages.azbuki.bg\/en\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/","title":{"rendered":"Apollonian Sphere and Properties of 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\/math2025-1-3-apo\" target=\"_blank\" rel=\"noopener\">https:\/\/doi.org\/10.53656\/math2025-1-3-apo<\/a><\/p>\n<p><strong>Abstract.<\/strong> The definition and some of the properties of the Apollonian circle in the plane find their analogies in the Euclidean three-dimensional space. Thus, we manage to introduce a new concept in solid geometry that we call an \u201cApollonian sphere\u201d. It appears that the Apollonian sphere not only possesses classical properties similar to the Apollonian circle such as orthogonality and coaxiality, but also analogies of its lesser-known connection with the Lemoine point and the circumcenter. We also discover two notable properties of stereographic projection that we prove with an Apollonian sphere. They include collinearity of the projection point with the Lemoine points of the projection and the projected triangles or with the centers of their Apollonian circles. Moreover, we connect the newly introduced concepts and the rich configurations they generate with Olympiad geometry.<br \/>\n<em>Keywords:<\/em> Apollonian sphere, stereographic projection, harmonic tetrahedron, Lemoine point, center of Apollonian circle, Olympiad geometry<\/p>\n<a href=\"https:\/\/foreignlanguages.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\/math2025-1-3-apo Abstract. The definition and some of the properties of the Apollonian circle in the plane find their analogies in the Euclidean three-dimensional space. Thus, we manage to introduce a new concept in solid geometry that we call an \u201cApollonian sphere\u201d. It appears that the Apollonian sphere not [&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":[10661,10665,10663,10664,10666,10662],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v21.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Apollonian Sphere and Properties of 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:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-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=\"Apollonian Sphere and Properties of 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\/math2025-1-3-apo Abstract. The definition and some of the properties of the Apollonian circle in the plane find their analogies in the Euclidean three-dimensional space. Thus, we manage to introduce a new concept in solid geometry that we call an \u201cApollonian sphere\u201d. It appears that the Apollonian sphere not [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-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=\"2025-04-04T08:49:28+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-09-17T11:28:09+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:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/\"},\"author\":{\"name\":\"\u201e\u0410\u0437-\u0431\u0443\u043a\u0438\u201c\",\"@id\":\"https:\/\/azbuki.bg\/en\/#\/schema\/person\/de220d282eaa494f914ce0fd838645dd\"},\"headline\":\"Apollonian Sphere and Properties of Stereographic Projection around the Lemoine Point\",\"datePublished\":\"2025-04-04T08:49:28+00:00\",\"dateModified\":\"2025-09-17T11:28:09+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/\"},\"wordCount\":200,\"publisher\":{\"@id\":\"https:\/\/azbuki.bg\/en\/#organization\"},\"keywords\":[\"Apollonian sphere\",\"center of Apollonian circle\",\"harmonic tetrahedron\",\"Lemoine point\",\"Olympiad geometry\",\"stereographic projection\"],\"inLanguage\":\"en-US\"},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/\",\"url\":\"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/\",\"name\":\"Apollonian Sphere and Properties of Stereographic Projection around the Lemoine Point - 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The definition and some of the properties of the Apollonian circle in the plane find their analogies in the Euclidean three-dimensional space. Thus, we manage to introduce a new concept in solid geometry that we call an \u201cApollonian sphere\u201d. It appears that the Apollonian sphere not [&hellip;]","og_url":"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/","og_site_name":"\u0410\u0437-\u0431\u0443\u043a\u0438","article_publisher":"https:\/\/www.facebook.com\/Azbuki55\/","article_published_time":"2025-04-04T08:49:28+00:00","article_modified_time":"2025-09-17T11:28:09+00:00","author":"\u201e\u0410\u0437-\u0431\u0443\u043a\u0438\u201c","twitter_card":"summary_large_image","twitter_misc":{"Written by":"\u201e\u0410\u0437-\u0431\u0443\u043a\u0438\u201c","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/#article","isPartOf":{"@id":"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/"},"author":{"name":"\u201e\u0410\u0437-\u0431\u0443\u043a\u0438\u201c","@id":"https:\/\/azbuki.bg\/en\/#\/schema\/person\/de220d282eaa494f914ce0fd838645dd"},"headline":"Apollonian Sphere and Properties of Stereographic Projection around the Lemoine Point","datePublished":"2025-04-04T08:49:28+00:00","dateModified":"2025-09-17T11:28:09+00:00","mainEntityOfPage":{"@id":"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/"},"wordCount":200,"publisher":{"@id":"https:\/\/azbuki.bg\/en\/#organization"},"keywords":["Apollonian sphere","center of Apollonian circle","harmonic tetrahedron","Lemoine point","Olympiad geometry","stereographic projection"],"inLanguage":"en-US"},{"@type":"WebPage","@id":"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/","url":"https:\/\/azbuki.bg\/uncategorized\/apollonian-sphere-and-properties-of-stereographic-projection-around-the-lemoine-point\/","name":"Apollonian Sphere and Properties of Stereographic Projection around the Lemoine Point - 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