https://artofproblemsolving.com/wiki/api.php?action=feedcontributions&user=LilliantheGeek&feedformat=atom AoPS Wiki - User contributions [en] 2021-09-22T06:33:57Z User contributions MediaWiki 1.31.1 https://artofproblemsolving.com/wiki/index.php?title=Carnot%27s_Theorem&diff=88868 Carnot's Theorem 2017-12-11T23:29:31Z <p>LilliantheGeek: /*Carnot's Theorem */</p> <hr /> <div>'''Carnot's Theorem''' states that in a [[triangle]] &lt;math&gt;ABC&lt;/math&gt;, the signed sum of [[perpendicular]] distances from the [[circumcenter]] &lt;math&gt;O&lt;/math&gt; to the sides (i.e., signed lengths of the pedal lines from &lt;math&gt;O&lt;/math&gt;) is:<br /> <br /> &lt;math&gt;OO_A+OO_B+OO_C=R+r&lt;/math&gt;<br /> <br /> &lt;asy&gt;<br /> pair a,b,c,O,i,d,f,g;<br /> a=(0,0);<br /> b=(4,0);<br /> c=(1,3);<br /> O=circumcenter(a,b,c);<br /> i=incenter(a,b,c);<br /> draw(a--b--c--cycle);<br /> draw(circumcircle(a,b,c));<br /> draw(incircle(a,b,c));<br /> dot(i);<br /> dot(O);<br /> label(&quot;$A$&quot;,a,W);<br /> label(&quot;$B$&quot;,b,E);<br /> label(&quot;$C$&quot;,c,N);<br /> label(&quot;$I$&quot;,i,N);<br /> label(&quot;$O$&quot;,O,N);<br /> d=foot(O,b,c);<br /> dot(d);<br /> draw(O--d);<br /> label(&quot;$O_A$&quot;,d,N);<br /> draw(rightanglemark(O,d,b));<br /> f=foot(O,a,b);<br /> dot(f);<br /> draw(O--f);<br /> draw(rightanglemark(O,f,a));<br /> label(&quot;$O_C$&quot;,f,S);<br /> g=foot(O,c,a);<br /> dot(g);<br /> draw(O--g);<br /> draw(rightanglemark(O,g,a));<br /> label(&quot;$O_B$&quot;,g,W);<br /> &lt;/asy&gt;<br /> <br /> where r is the [[inradius]] and R is the [[circumradius]]. The sign of the distance is chosen to be negative iff the entire segment OO_i lies outside the triangle.<br /> Explicitly,<br /> <br /> &lt;math&gt;OO_A+OO_B+OO_C=\frac{abc(|\cos{A}|+|\cos{B}|+|\cos{C}|)}{4|\Delta|}&lt;/math&gt;<br /> <br /> where &lt;math&gt;\Delta&lt;/math&gt; is the area of triangle &lt;math&gt;\Delta ABC&lt;/math&gt;.<br /> <br /> <br /> Weisstein, Eric W. &quot;Carnot's Theorem.&quot; From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/CarnotsTheorem.html<br /> <br /> <br /> =Carnot's Theorem=<br /> <br /> '''Carnot's Theorem''' states that in a [[triangle]] &lt;math&gt;ABC&lt;/math&gt; with &lt;math&gt;A_1\in BC&lt;/math&gt;, &lt;math&gt;B_1\in AC&lt;/math&gt;, and &lt;math&gt;C_1\in AB&lt;/math&gt;, [[perpendicular]]s to the sides &lt;math&gt;BC&lt;/math&gt;, &lt;math&gt;AC&lt;/math&gt;, and &lt;math&gt;AB&lt;/math&gt; at &lt;math&gt;A_1&lt;/math&gt;, &lt;math&gt;B_1&lt;/math&gt;, and &lt;math&gt;C_1&lt;/math&gt; are [[concurrent]] [[iff|if and only if]] &lt;math&gt;A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2&lt;/math&gt;.<br /> <br /> ====Proof====<br /> '''Only if:''' Assume that the given perpendiculars are concurrent at &lt;math&gt;M&lt;/math&gt;. Then, from the Pythagorean Theorem, &lt;math&gt;A_1B^2=BM^2-MA_1^2&lt;/math&gt;, &lt;math&gt;C_1A^2=AM^2-MC_1^2&lt;/math&gt;, &lt;math&gt;B_1C^2=CM^2-MB_1^2&lt;/math&gt;, &lt;math&gt;A_1C^2=MC^2-MA_1^2&lt;/math&gt;, &lt;math&gt;C_1B^2=MB^2-MC_1^2&lt;/math&gt;, and &lt;math&gt;B_1A^2=AM^2-MB_1^2&lt;/math&gt;. Substituting each and every one of these in and simplifying gives the desired result.<br /> <br /> <br /> <br /> ''' If:''' Consider the intersection of the perpendiculars from &lt;math&gt;A_1&lt;/math&gt; and &lt;math&gt;B_1&lt;/math&gt;. Call this intersection point &lt;math&gt;N&lt;/math&gt;, and let &lt;math&gt;C_2&lt;/math&gt; be the perpendicular from &lt;math&gt;N&lt;/math&gt; to &lt;math&gt;AB&lt;/math&gt;. From the other direction of the desired result, we have that &lt;math&gt;A_1B^2+C_2A^2+B_1C^2=A_1C^2+C_2B^2+B_1A^2&lt;/math&gt;. We also have that &lt;math&gt;A_1B^2+C_1A^2+B_1C^2=A_1C^2+C_1B^2+B_1A^2&lt;/math&gt;, which implies that &lt;math&gt;C_1A^2-C_1B^2=C_2A^2-C_2B^2&lt;/math&gt;. This is a difference of squares, which we can easily factor into &lt;math&gt;(C_1A-C_1B)(C_1A+C_1B)=(C_2A-C_2B)(C_2A+C_2B)&lt;/math&gt;. Note that &lt;math&gt;C_1A+C_1=C_2A+C_2B=AB&lt;/math&gt;, so we have that &lt;math&gt;C_1A-C_1B=C_2A-C_2B&lt;/math&gt;. This implies that &lt;math&gt;C_1=C_2&lt;/math&gt;, which gives the desired result.<br /> <br /> ====Problems====<br /> ===Olympiad===<br /> &lt;math&gt;\triangle ABC&lt;/math&gt; is a triangle. Take points &lt;math&gt;D, E, F&lt;/math&gt; on the perpendicular bisectors of &lt;math&gt;BC, CA, AB&lt;/math&gt; respectively. Show that the lines through &lt;math&gt;A, B, C&lt;/math&gt; perpendicular to &lt;math&gt;EF, FD, DE&lt;/math&gt; respectively are concurrent. ([[1997 USAMO Problems/Problem 2|Source]])<br /> <br /> ==See also==<br /> <br /> [[Carnot's Polygon Theorem]]<br /> [[Japanese Theorem]]<br /> [[Category:Geometry]]<br /> [[Category:Theorems]]</div> LilliantheGeek https://artofproblemsolving.com/wiki/index.php?title=Gmaas&diff=79670 Gmaas 2016-07-22T00:46:45Z <p>LilliantheGeek: /* Known Facts About gmaas */</p> <hr /> <div>=== Known Facts About gmaas ===<br /> <br /> - Gmaas is 5space's favorite animal. [http://artofproblemsolving.com/wiki/index.php?title=File:Gmaas2.png (Source)]<br /> <br /> - He lives with sseraj. <br /> <br /> - He has 20 supercars, including a new cat-themed Ferrari 458 Spider that he dubbed the Purrari.<br /> [img]https://i.ytimg.com/vi/8JKeYy4X3KE/maxresdefault.jpg[/img]<br /> <br /> - He is often overfed by sseraj.<br /> <br /> - He is an employee of AoPS and teaches but doesn't get paid.<br /> <br /> - He is a gmaas with yellow fur and white hypnotizing eyes.<br /> <br /> - He was born with a tail that is a completely different color from the rest of his fur.<br /> <br /> - His stare is very hypnotizing and effective at getting table scraps.<br /> <br /> - He sometimes appears about several thousand hours before certain classes (but it can vary), such as Introduction to Algebra B, as an admin. <br /> <br /> - He died from too many Rubik's cubes in an Introduction to Algebra A class, but got revived by the Dark Lord at 3:15 AM the next day.<br /> <br /> - It is uncertain whether or not he is a cat, or is merely some sort of beast that has chosen to take the form of a cat (specifically a Persian Smoke.) <br /> <br /> - Actually, he is a cat. He said so.<br /> <br /> - He is very famous now, and mods always talk about him before class starts.<br /> <br /> - His favorite food is AoPS textbooks, because they help him &lt;math&gt;[i]digest[\i]&lt;/math&gt; problems.<br /> <br /> - Gmaas tends to reside in sseraj's fridge.<br /> <br /> - Gmaas once ate all sseraj's fridge food, so sseraj had to put him in the freezer.<br /> <br /> - The fur of Gmaas can protect him from the harsh conditions of a freezer.<br /> <br /> - Gmaas sightings are not very common. There have only been 8 confirmed sightings of Gmaas in the wild.<br /> <br /> -Gmaas is a sage omniscient cat.<br /> <br /> -He is looking for suitable places other than sseraj's fridge to live in.<br /> <br /> - Places where gmaas sightings have happened: <br /> ~MouseFeastForCats/CAT 8 Mouse Apartment 1083<br /> ~Alligator Swamp A 1072 <br /> ~Alligator Swamp B 1073<br /> ~Introduction to Algebra A (1170)<br /> ~Welcome to Panda Town Gate 1076<br /> ~Welcome to Gmaas Town Gate 1221<br /> ~Welcome to Gmaas Town Gate 1125<br /> ~33°01'17.4&quot;N 117°05'40.1&quot;W<br /> ~AoPS<br /> ~The other side of the ice in Antarctica<br /> ~Feisty Alligator Swamp 1115<br /> ~Introduction to Geometry 1221 (Taught by sseraj)<br /> ~Introduction to Counting and Probability 1142 <br /> ~Feisty-ish Alligator Swamp 1115 (AGAIN)<br /> ~Intermediate Counting and Probability 1137<br /> ~Intermediate Counting and Probability 1207<br /> ~Posting student surveys<br /> ~USF Castle Walls 1203<br /> ~Dark Lord's Hut 1210<br /> ~AMC 10 Problem Series 1200<br /> ~Intermediate Number Theory 1138<br /> ~Introduction To Number Theory 1204<br /> ~Algebra B 1112<br /> - These have all been designated as the most glorious sections of Aopsland now (except the USF castle walls), but deforestation threatens the wild areas (i.e. Alligator Swamps A&amp;B).<br /> <br /> - Gmaas has also been sighted in Olympiad Geometry 1148.<br /> <br /> - Gmaas are often under the disguise of a penguin or cat. Look out for them.<br /> <br /> - Gmaas is the master of possessing.<br /> <br /> - He lives in the shadows. Is he a dream? Truth? Fiction? Condemnation? Salvation? AoPS site admin? He is all these things and none of them. He is... Gmaas.<br /> <br /> - If you make yourself more than just a cat... if you devote yourself to an ideal... and if they can't stop you... then you become something else entirely. A LEGEND. Gmaas now belongs to the ages.<br /> <br /> - Is this the real life? Is this just fantasy? No. This is gmaas, the legend.<br /> <br /> - moab33 and sxu have both predicted (accurately) the behavior on egg124 and MinionLA, respectively. The classes were Intro to Prealgebra 1, coincidentally taught by sseraj, and Intro to Algebra (unknown part.) They may be have been possesed by Gmaas. Scientist moab33 is looking into it.<br /> <br /> - cobra has seen Gmaas viewing the Ultimate Survival Forum. He (or is he a she?) is suspected to be transforming the characters into real life. Be prepared to meet your epic swordsman self someday.<br /> <br /> === gmaas in Popular Culture ===<br /> <br /> - Currently, [https://docs.google.com/document/d/1mLa2d_9Qgv4C9cZdThyjA6kSf2ULgwvkVjPVqmsoV2w/edit a book] is being written (by JpusheenS) about the adventures of gmaas. It is aptly titled, &quot;The Adventures of gmaas&quot;.<br /> <br /> - BREAKING NEWS: tigershark22 has found a possible cousin to gmaas in Raymond Feist's book Silverthorn. They are mountain dwellers, gwali. Not much are known about them either, and when someone asked,&quot;What are gwali?&quot; the customary answer &quot;This is gwali&quot; is returned. Scientist 5space is now looking into it.<br /> <br /> - Sullymath is also writing a book about Gmaas<br /> <br /> - Potential sighting of gmass [http://www.gmac.com/frequently-asked-questions/gmass-search-service.aspx]<br /> <br /> - Gmaas has been spotted in some Doctor Who and Phineas and Ferb episodes, such as Aliens of London, Phineas and Ferb Save Summer, and many more.</div> LilliantheGeek