| Summary: | The notion of the Gergonne point of a triangle in the Euclidean plane is very well known, and the study of them in the isotropic setting has already appeared earlier. In this paper, we give two generalizations of the Gergonne point of a triangle in the isotropic plane, and we study several curves related to them. The first generalization is based on the fact that for the triangle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></semantics></math></inline-formula> and its contact triangle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>i</mi></msub><msub><mi>B</mi><mi>i</mi></msub><msub><mi>C</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, there is a pencil of circles such that each circle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>k</mi><mi>m</mi></msub></semantics></math></inline-formula> from the pencil the lines <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><msub><mi>A</mi><mi>m</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>B</mi><msub><mi>B</mi><mi>m</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>C</mi><mi>m</mi></msub></mrow></semantics></math></inline-formula> is concurrent at a point <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mi>m</mi></msub></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>A</mi><mi>m</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>B</mi><mi>m</mi></msub></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>C</mi><mi>m</mi></msub></semantics></math></inline-formula> are points on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>k</mi><mi>m</mi></msub></semantics></math></inline-formula> parallel to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mi>i</mi></msub><mo>,</mo><msub><mi>B</mi><mi>i</mi></msub><mo>,</mo><msub><mi>C</mi><mi>i</mi></msub></mrow></semantics></math></inline-formula>, respectively. To introduce the second generalization of the Gergonne point, we prove that for the triangle <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mi>B</mi><mi>C</mi></mrow></semantics></math></inline-formula>, point <i>I</i> and three lines <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>,</mo><msub><mi>q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>q</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> through <i>I</i> there are two points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula> such that for the points <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Q</mi><mn>1</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>Q</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>q</mi><mn>1</mn></msub><mo>,</mo><msub><mi>q</mi><mn>2</mn></msub><mo>,</mo><msub><mi>q</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>d</mi><mrow><mo>(</mo><mi>I</mi><mo>,</mo><msub><mi>Q</mi><mn>1</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>d</mi><mrow><mo>(</mo><mi>I</mi><mo>,</mo><msub><mi>Q</mi><mn>2</mn></msub><mo>)</mo></mrow><mo>=</mo><mi>d</mi><mrow><mo>(</mo><mi>I</mi><mo>,</mo><msub><mi>Q</mi><mn>3</mn></msub><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, the lines <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><msub><mi>Q</mi><mn>1</mn></msub><mo>,</mo><mi>B</mi><msub><mi>Q</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>C</mi><msub><mi>Q</mi><mn>3</mn></msub></mrow></semantics></math></inline-formula> are concurrent at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>G</mi><mrow><mn>1</mn><mo>,</mo><mn>2</mn></mrow></msub></semantics></math></inline-formula>. We achieve these results by using the standardization of the triangle in the isotropic plane and simple analytical method.
|
|---|