https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=JeriC&feedformat=atomAoPS Wiki - User contributions [en]2020-10-21T06:46:00ZUser contributionsMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=11012Euler line2006-11-05T15:46:49Z<p>JeriC: </p>
<hr />
<div>{{stub}}<br />
<br />
Let <math>ABC</math> be a triangle, points <math>H, N, G, O, L</math> as <math>\triangle ABC</math>'s [[orthocenter]], [[nine-point center]], [[centroid]], [[circumcenter]], [[De Longchamps point]] respectively, then these points are [[collinear]](regardless of <math>\triangle ABC</math>'s shape). And the line passes through points <math>H, N, G, O, L</math> 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 />
* <math>OG:GN:NH\equiv2:1:3</math><br />
<br />
Construct an [[orthic triangle]]<math>\triangle H_AH_BH_C</math>, then Euler lines of <math>\triangle AH_BH_C</math>,<math>\triangle BH_CH_A</math>,<math>\triangle CH_AH_B</math> concur at <math>\triangle ABC</math>'s [[nine-point center]].</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Nine_point_center&diff=10963Nine point center2006-11-05T02:06:23Z<p>JeriC: </p>
<hr />
<div>The [[center]] of the [[Nine point circle]].</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Collinear&diff=10959Collinear2006-11-05T01:37:28Z<p>JeriC: </p>
<hr />
<div>'''Collinear''' means "lying on the same [[line]]." Thus, any two [[point]]s on a [[plane]] are collinear. <br />
<br />
==See also==<br />
[[Menelaus' Theorem]]<br />
<br />
[[Category:Definition]]</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=10958Euler line2006-11-05T01:27:05Z<p>JeriC: </p>
<hr />
<div>{{stub}}<br />
<br />
Let <math>ABC</math> be a triangle, points <math>H, N, G, O, L</math> as <math>\triangle ABC</math>'s [[orthocenter]], [[nine-point center]], [[centroid]], [[circumcenter]], [[De Longchamps point]] respectively, then these points are collinear(regardless of <math>\triangle ABC</math>'s shape). And the line passes through points <math>H, N, G, O, L</math> 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 />
* <math>OG:GN:NH\equiv2:1:3</math><br />
<br />
Construct an [[orthic triangle]]<math>\triangle H_A,H_B,H_C</math>, then Euler lines of <math>\triangle AH_BH_C</math>,<math>\triangle BH_CH_A</math>,<math>\triangle CH_AH_B</math> concur at <math>\triangle ABC</math>'s [[nine-point center]].</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Fermat_point&diff=10916Fermat point2006-11-05T00:40:32Z<p>JeriC: </p>
<hr />
<div>{{stub}}<br />
<br />
Also called '''Torricelli point'''.<br />
<br />
In a triangle <math>\triangle ABC</math>, a point <math>p</math> which has the minimum total distance to three [[vertices]]. (i.e., <math>|Ap|+|Bp|+|Cp|)</math> 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 <math>\triangle ABC</math>, then connect each new vertex to each opposite vertex, as these three lines will concur at first Fermat point.</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Fermat_point&diff=10915Fermat point2006-11-05T00:40:17Z<p>JeriC: </p>
<hr />
<div>{{stub}}<br />
<br />
Also called '''Torricelli point'''.<br />
<br />
In atriangle <math>\triangle ABC</math>, a point <math>p</math> which has the minimum total distance to three [[vertices]]. (i.e., <math>|Ap|+|Bp|+|Cp|)</math> 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 <math>\triangle ABC</math>, then connect each new vertex to each opposite vertex, as these three lines will concur at first Fermat point.</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10883Talk:Euler line2006-11-05T00:06:02Z<p>JeriC: </p>
<hr />
<div>Should "Euler's line" redirect here or should "Euler line" redirect to "Euler's line"? 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 "The Euler line of triangle <math>\displaystyle ABC</math>" sounds much less silly than "The Euler's line of triangle <math>\displaystyle ABC</math>". But in any event, if you rename an article, you should rename its associated discussion section as well. &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 "the Euler line of a triangle." I've moved the content here; "Euler's line" is now a redirect.--[[User:JBL|JBL]] 19:00, 4 November 2006 (EST)</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10881Talk:Euler line2006-11-05T00:05:04Z<p>JeriC: </p>
<hr />
<div>Should "Euler's line" redirect here or should "Euler line" redirect to "Euler's line"? 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 "The Euler line of triangle <math>\displaystyle ABC</math>" sounds much less silly than "The Euler's line of triangle <math>\displaystyle ABC</math>". But in any event, if you rename an article, you should rename its associated discussion section as well. &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 "the Euler line of a triangle." I've moved the content here; "Euler's line" 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>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Nine-point_circle&diff=10838Nine-point circle2006-11-04T23:17:18Z<p>JeriC: </p>
<hr />
<div>#REDIRECT[[Nine point circle]]</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Nine-point_circle&diff=10836Nine-point circle2006-11-04T23:17:02Z<p>JeriC: </p>
<hr />
<div>#REDIRECT Nine point circle</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Euler_point&diff=10833Euler point2006-11-04T23:16:05Z<p>JeriC: </p>
<hr />
<div>The Euler points, usually denoted as <math>E_A,E_B,E_C</math>, are the midpoints of the orthocenter of a given triangle<math>\triangle ABC</math> to the vertex <math>A,B, and C</math>.<br />
<br />
Euler points lie on the [[Nine-point circle]].</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Nine_point_circle&diff=10831Nine point circle2006-11-04T23:14:55Z<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 <math>\triangle ABC</math>'s feet of altitude dropped from three vertices <math>A,B</math>, and <math>C</math>, usually denoted as <math>H_A,H_B,H_C</math>, and midpoints of three sides, as <math>M_A,M_B,M_C</math>, and three [[Euler point]]s <math>E_A,E_B,E_C</math>.</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Euler_point&diff=10814Euler point2006-11-04T22:39:48Z<p>JeriC: </p>
<hr />
<div>The Euler points, usually denoted as <math>E_A,E_B,E_C</math>, are the midpoints of the orthocenter of a given triangle<math>\triangle ABC</math> to the vertex <math>A,B, and C</math>.<br />
<br />
Euler points lie on the [[nine-point circle]].</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10800Talk:Euler line2006-11-04T21:53:08Z<p>JeriC: </p>
<hr />
<div>Should "Euler's line" redirect here or should "Euler line" redirect to "Euler's line"? 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>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Talk:Euler_line&diff=10799Talk:Euler line2006-11-04T21:52:16Z<p>JeriC: </p>
<hr />
<div>Should "Euler's line" redirect here or should "Euler line" redirect to "Euler's line"? 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>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=10798Euler line2006-11-04T21:51:22Z<p>JeriC: </p>
<hr />
<div>{{stub}}<br />
<br />
Let <math>ABC</math> be a triangle, points <math>H, N, G, O, L</math> as <math>\triangle ABC</math>'s [[orthocenter]], [[nine-point center]], [[centroid]], [[circumcenter]], [[De Longchamps point]] respectively, then these points are collinear(regardless of <math>\triangle ABC</math>'s shape). And the line passes through points <math>H, N, G, O, L</math> 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 />
* <math>OG:GN:NH\equiv2:1:3</math></div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Euler_line&diff=10783Euler line2006-11-04T21:02:51Z<p>JeriC: </p>
<hr />
<div>#REDIRECT[[Euler line]]</div>JeriChttps://artofproblemsolving.com/wiki/index.php?title=Leonhard_Euler&diff=10756Leonhard Euler2006-11-04T18:17:36Z<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 <math>\zeta(2) = \frac{\pi^2}6</math> where <math>\zeta</math> 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