https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=JeriC&feedformat=atom AoPS Wiki - User contributions [en] 2020-10-21T06:46:00Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=11012 Euler line 2006-11-05T15:46:49Z <p>JeriC: </p> <hr /> <div>{{stub}}<br /> <br /> Let &lt;math&gt;ABC&lt;/math&gt; be a triangle, points &lt;math&gt;H, N, G, O, L&lt;/math&gt; as &lt;math&gt;\triangle ABC&lt;/math&gt;'s [[orthocenter]], [[nine-point center]], [[centroid]], [[circumcenter]], [[De Longchamps point]] respectively, then these points are [[collinear]](regardless of &lt;math&gt;\triangle ABC&lt;/math&gt;'s shape). And the line passes through points &lt;math&gt;H, N, G, O, L&lt;/math&gt; is called as Euler line, which is named after [[Leonhard Euler]].<br /> <br /> An interesting property of distances between these points on the Euler line:<br /> * &lt;math&gt;OG:GN:NH\equiv2:1:3&lt;/math&gt;<br /> <br /> Construct an [[orthic triangle]]&lt;math&gt;\triangle H_AH_BH_C&lt;/math&gt;, then Euler lines of &lt;math&gt;\triangle AH_BH_C&lt;/math&gt;,&lt;math&gt;\triangle BH_CH_A&lt;/math&gt;,&lt;math&gt;\triangle CH_AH_B&lt;/math&gt; concur at &lt;math&gt;\triangle ABC&lt;/math&gt;'s [[nine-point center]].</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Nine_point_center&diff=10963 Nine point center 2006-11-05T02:06:23Z <p>JeriC: </p> <hr /> <div>The [[center]] of the [[Nine point circle]].</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Collinear&diff=10959 Collinear 2006-11-05T01:37:28Z <p>JeriC: </p> <hr /> <div>'''Collinear''' means &quot;lying on the same [[line]].&quot; Thus, any two [[point]]s on a [[plane]] are collinear. <br /> <br /> ==See also==<br /> [[Menelaus' Theorem]]<br /> <br /> [[Category:Definition]]</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=10958 Euler line 2006-11-05T01:27:05Z <p>JeriC: </p> <hr /> <div>{{stub}}<br /> <br /> Let &lt;math&gt;ABC&lt;/math&gt; be a triangle, points &lt;math&gt;H, N, G, O, L&lt;/math&gt; as &lt;math&gt;\triangle ABC&lt;/math&gt;'s [[orthocenter]], [[nine-point center]], [[centroid]], [[circumcenter]], [[De Longchamps point]] respectively, then these points are collinear(regardless of &lt;math&gt;\triangle ABC&lt;/math&gt;'s shape). And the line passes through points &lt;math&gt;H, N, G, O, L&lt;/math&gt; is called as Euler line, which is named after [[Leonhard Euler]].<br /> <br /> An interesting property of distances between these points on the Euler line:<br /> * &lt;math&gt;OG:GN:NH\equiv2:1:3&lt;/math&gt;<br /> <br /> Construct an [[orthic triangle]]&lt;math&gt;\triangle H_A,H_B,H_C&lt;/math&gt;, then Euler lines of &lt;math&gt;\triangle AH_BH_C&lt;/math&gt;,&lt;math&gt;\triangle BH_CH_A&lt;/math&gt;,&lt;math&gt;\triangle CH_AH_B&lt;/math&gt; concur at &lt;math&gt;\triangle ABC&lt;/math&gt;'s [[nine-point center]].</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Fermat_point&diff=10916 Fermat point 2006-11-05T00:40:32Z <p>JeriC: </p> <hr /> <div>{{stub}}<br /> <br /> Also called '''Torricelli point'''.<br /> <br /> In a triangle &lt;math&gt;\triangle ABC&lt;/math&gt;, a point &lt;math&gt;p&lt;/math&gt; which has the minimum total distance to three [[vertices]]. (i.e., &lt;math&gt;|Ap|+|Bp|+|Cp|)&lt;/math&gt; is called the first Fermat point or simply '''Fermat point''' in general.<br /> <br /> A method to find the point is to construct three equilateral triangles out of the three sides from &lt;math&gt;\triangle ABC&lt;/math&gt;, then connect each new vertex to each opposite vertex, as these three lines will concur at first Fermat point.</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Fermat_point&diff=10915 Fermat point 2006-11-05T00:40:17Z <p>JeriC: </p> <hr /> <div>{{stub}}<br /> <br /> Also called '''Torricelli point'''.<br /> <br /> In atriangle &lt;math&gt;\triangle ABC&lt;/math&gt;, a point &lt;math&gt;p&lt;/math&gt; which has the minimum total distance to three [[vertices]]. (i.e., &lt;math&gt;|Ap|+|Bp|+|Cp|)&lt;/math&gt; is called the first Fermat point or simply '''Fermat point''' in general.<br /> <br /> A method to find the point is to construct three equilateral triangles out of the three sides from &lt;math&gt;\triangle ABC&lt;/math&gt;, then connect each new vertex to each opposite vertex, as these three lines will concur at first Fermat point.</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10883 Talk:Euler line 2006-11-05T00:06:02Z <p>JeriC: </p> <hr /> <div>Should &quot;Euler's line&quot; redirect here or should &quot;Euler line&quot; redirect to &quot;Euler's line&quot;? I think the second is better, because it's ''his'' line. --[[User:I like pie|I_like_pie]] 16:46, 4 November 2006 (EST)<br /> <br /> Edited :) -- JeriC<br /> <br /> Actually, I prefer Euler line. (I think it's referenced as such in most published materials anyhow, but I can't support my claim.) I also think that it in general, it is usual to name geometric objects after their discoverers without using possessives (e.g., the Simson line, Brocard point, Fermat point). I think that the reason for this might be that &quot;The Euler line of triangle &lt;math&gt;\displaystyle ABC&lt;/math&gt;&quot; sounds much less silly than &quot;The Euler's line of triangle &lt;math&gt;\displaystyle ABC&lt;/math&gt;&quot;. But in any event, if you rename an article, you should rename its associated discussion section as well. &amp;mdash;[[User:Boy Soprano II|Boy Soprano II]] 18:50, 4 November 2006 (EST)<br /> <br /> Good point... So, what do we do? I think I'll change my mind and say you're right. --[[User:I like pie|I_like_pie]] 18:51, 4 November 2006 (EST)<br /> <br /> It's &quot;the Euler line of a triangle.&quot; I've moved the content here; &quot;Euler's line&quot; is now a redirect.--[[User:JBL|JBL]] 19:00, 4 November 2006 (EST)</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10881 Talk:Euler line 2006-11-05T00:05:04Z <p>JeriC: </p> <hr /> <div>Should &quot;Euler's line&quot; redirect here or should &quot;Euler line&quot; redirect to &quot;Euler's line&quot;? I think the second is better, because it's ''his'' line. --[[User:I like pie|I_like_pie]] 16:46, 4 November 2006 (EST)<br /> <br /> Edited :) -- JeriC<br /> <br /> Actually, I prefer Euler line. (I think it's referenced as such in most published materials anyhow, but I can't support my claim.) I also think that it in general, it is usual to name geometric objects after their discoverers without using possessives (e.g., the Simson line, Brocard point, Fermat point). I think that the reason for this might be that &quot;The Euler line of triangle &lt;math&gt;\displaystyle ABC&lt;/math&gt;&quot; sounds much less silly than &quot;The Euler's line of triangle &lt;math&gt;\displaystyle ABC&lt;/math&gt;&quot;. But in any event, if you rename an article, you should rename its associated discussion section as well. &amp;mdash;[[User:Boy Soprano II|Boy Soprano II]] 18:50, 4 November 2006 (EST)<br /> <br /> Good point... So, what do we do? I think I'll change my mind and say you're right. --[[User:I like pie|I_like_pie]] 18:51, 4 November 2006 (EST)<br /> <br /> It's &quot;the Euler line of a triangle.&quot; I've moved the content here; &quot;Euler's line&quot; is now a redirect.--[[User:JBL|JBL]] 19:00, 4 November 2006 (EST)<br /> <br /> Maybe we can just let two pages redirecting each other so that everyone is happy. ;) --JeriC</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Nine-point_circle&diff=10838 Nine-point circle 2006-11-04T23:17:18Z <p>JeriC: </p> <hr /> <div>#REDIRECT[[Nine point circle]]</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Nine-point_circle&diff=10836 Nine-point circle 2006-11-04T23:17:02Z <p>JeriC: </p> <hr /> <div>#REDIRECT Nine point circle</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Euler_point&diff=10833 Euler point 2006-11-04T23:16:05Z <p>JeriC: </p> <hr /> <div>The Euler points, usually denoted as &lt;math&gt;E_A,E_B,E_C&lt;/math&gt;, are the midpoints of the orthocenter of a given triangle&lt;math&gt;\triangle ABC&lt;/math&gt; to the vertex &lt;math&gt;A,B, and C&lt;/math&gt;.<br /> <br /> Euler points lie on the [[Nine-point circle]].</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Nine_point_circle&diff=10831 Nine point circle 2006-11-04T23:14:55Z <p>JeriC: </p> <hr /> <div>Also known as Euler's circle or Feuerbach's circle, as its name introduces itself that the nine point circle passes nine points, which are a given triangle &lt;math&gt;\triangle ABC&lt;/math&gt;'s feet of altitude dropped from three vertices &lt;math&gt;A,B&lt;/math&gt;, and &lt;math&gt;C&lt;/math&gt;, usually denoted as &lt;math&gt;H_A,H_B,H_C&lt;/math&gt;, and midpoints of three sides, as &lt;math&gt;M_A,M_B,M_C&lt;/math&gt;, and three [[Euler point]]s &lt;math&gt;E_A,E_B,E_C&lt;/math&gt;.</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Euler_point&diff=10814 Euler point 2006-11-04T22:39:48Z <p>JeriC: </p> <hr /> <div>The Euler points, usually denoted as &lt;math&gt;E_A,E_B,E_C&lt;/math&gt;, are the midpoints of the orthocenter of a given triangle&lt;math&gt;\triangle ABC&lt;/math&gt; to the vertex &lt;math&gt;A,B, and C&lt;/math&gt;.<br /> <br /> Euler points lie on the [[nine-point circle]].</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10800 Talk:Euler line 2006-11-04T21:53:08Z <p>JeriC: </p> <hr /> <div>Should &quot;Euler's line&quot; redirect here or should &quot;Euler line&quot; redirect to &quot;Euler's line&quot;? I think the second is better, because it's ''his'' line. --[[User:I like pie|I_like_pie]] 16:46, 4 November 2006 (EST)<br /> <br /> Edited :) -- JeriC</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10799 Talk:Euler line 2006-11-04T21:52:16Z <p>JeriC: </p> <hr /> <div>Should &quot;Euler's line&quot; redirect here or should &quot;Euler line&quot; redirect to &quot;Euler's line&quot;? I think the second is better, because it's ''his'' line. --[[User:I like pie|I_like_pie]] 16:46, 4 November 2006 (EST)<br /> <br /> Edited :)<br /> ----</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=10798 Euler line 2006-11-04T21:51:22Z <p>JeriC: </p> <hr /> <div>{{stub}}<br /> <br /> Let &lt;math&gt;ABC&lt;/math&gt; be a triangle, points &lt;math&gt;H, N, G, O, L&lt;/math&gt; as &lt;math&gt;\triangle ABC&lt;/math&gt;'s [[orthocenter]], [[nine-point center]], [[centroid]], [[circumcenter]], [[De Longchamps point]] respectively, then these points are collinear(regardless of &lt;math&gt;\triangle ABC&lt;/math&gt;'s shape). And the line passes through points &lt;math&gt;H, N, G, O, L&lt;/math&gt; is called as Euler line, which is named after [[Leonhard Euler]].<br /> <br /> An interesting property of distances between these points on the Euler line:<br /> * &lt;math&gt;OG:GN:NH\equiv2:1:3&lt;/math&gt;</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=10783 Euler line 2006-11-04T21:02:51Z <p>JeriC: </p> <hr /> <div>#REDIRECT[[Euler line]]</div> JeriC https://artofproblemsolving.com/wiki/index.php?title=Leonhard_Euler&diff=10756 Leonhard Euler 2006-11-04T18:17:36Z <p>JeriC: /* Biography */</p> <hr /> <div>{{stub}}<br /> <br /> '''Leonhard Euler''' (pronounced ''Oiler'') was a famous Swiss [[mathematician]]. He made numerous contributions to many fields of [[mathematics]] and [[science]]. Euler is considered to be one of the greatest mathematicians of all time.<br /> <br /> == Biography ==<br /> Euler was born on April 15, 1707 in Basel, Switzerland. Euler's parents were Paul Euler, a pastor of the Reformed Church, and Marguerite Brucker, a pastor's daughter. He had two young sisters, named Anna Maria and Maria Magdalena. At the age of thirteen he enrolled at the [[University of Basel]].<br /> <br /> On January 7, 1734, he married Katharina Gsell. The young couple had thirteen children, only five of whom survived childhood.<br /> <br /> In 1735, Euler was blind in his right eye and 1771 the left, however it obstructed nearly none of his prolific research productivity. <br /> <br /> ''More information needed.''<br /> <br /> On September 18, 1783, Euler passed away in St. Petersburg, Russia after suffering a brain hemorrhage. He was buried in the Alexander Nevsky Monastery.<br /> <br /> == Contributions ==<br /> * In 1735, Euler showed that &lt;math&gt;\zeta(2) = \frac{\pi^2}6&lt;/math&gt; where &lt;math&gt;\zeta&lt;/math&gt; is the [[zeta function]].<br /> * [[Euler's polyhedral formula]]<br /> * [[Euler's totient theorem]]<br /> * [[Euler's identity]]<br /> * [[Eulerian graph]]<br /> <br /> == See Also ==<br /> * [[Euler's line]]<br /> <br /> [[Category:Famous Mathematicians]]</div> JeriC