tag:blogger.com,1999:blog-6933544261975483399.post9169531321710731887..comments2024-03-26T19:10:02.918-07:00Comments on Go Geometry (Problem Solutions): Geometry Problem 1146: Triangle, Interior Point, 10, 20, 30, 130 degrees, CongruenceAntonio Gutierrezhttp://www.blogger.com/profile/04521650748152459860noreply@blogger.comBlogger21125tag:blogger.com,1999:blog-6933544261975483399.post-57996545095619333212021-06-02T22:44:43.586-07:002021-06-02T22:44:43.586-07:00Hi
How circumcenter of tr. ACP?Hi<br /><br />How circumcenter of tr. ACP?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-88199092361517057202021-06-02T22:40:33.768-07:002021-06-02T22:40:33.768-07:00Hi
How D is the circumcenter of tr. ACP?Hi <br /><br />How D is the circumcenter of tr. ACP?kasep10https://www.blogger.com/profile/09187154747362214621noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-76465805316988779952020-10-12T14:20:39.134-07:002020-10-12T14:20:39.134-07:00Extend AE to a point G such that tr AGC is a right...Extend AE to a point G such that tr AGC is a right triangle. The right triangle AGC obtained is congruent to tr CFO. Also triangle EGC is 30-60-90.<br />This helps with CE=AFAnonymoushttps://www.blogger.com/profile/17821589001449105440noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-46444777433026275732020-05-02T19:48:35.177-07:002020-05-02T19:48:35.177-07:00In fact, it is a special case of more general prob...In fact, it is a special case of more general problem.<br />t=10; (t, 30-2t):(2*t, 30-2t) -> (90+t, 30)<br />See https://output.jsbin.com/fofecum#20,10,10,10.<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-37571877374246127142016-08-23T04:07:52.628-07:002016-08-23T04:07:52.628-07:00"Read Mind Map on this site."
Which one?..."Read Mind Map on this site."<br />Which one? You mean this one?:<br />http://www.gogeometry.com/problem/problem_solving_techniques_mind_map.html<br />I read it, but it didn't much informative on the subject of solving geometry problems, because it's too general. I'd rather like to know how to figure out what auxiliary lines I need to draw and where to make the problem solvable.<br /><br />_"What we see is the end result of the Problem often much refined."_<br />That's exactly what I have problem with, because such an result doesn't contain the path which lead to that particular solution. And I'm more interested in how can I come up with a solution myself instead of just staring in awe at how others did it and puzzling how did they figure it out.<br /><br />_"The process of course is mainly trial and error"_<br />In my case, my every trial so far ends up with me being stuck somewhere and not knowing how to progress any further :/ There's always something missing between where I am and the statement to prove. (Even if I try going backwards from the last statement.)<br /><br />I'd like to learn how to solve such problems, because I stumble at many problems in my own research in geometry where I need to calculate angles and lengths in different polygons, and it always take me hours of hopeless brooding until I find a way to calculate something I need. Learning these techniques could save me a lot of time.<br /><br />_"Try to infer whatever u can from the given data, some or all of these inferences may lead u to the answer."_<br />That's what I do. But I always get stuck somewhere eventually.<br /><br />_"Many a time these problems can be deviously difficult or most obvious."_<br />Is there any list of the problems on this website arranged by the level of difficulty? I'd like to start from something simple and learn gradually.<br />Oh, and how to see if a problem has been solved or not? Are there listed somewhere on separate lists? (solved vs. unsolved)<br />Sorry if my questions are dumb, but I find this website to be a real mess, lots and lots of subpages and I can't really grasp its "structure" :/<br /><br />"_Let me explain my thought process on Antonio's latest Problem (1250) which appeared today in Gogeometry."_<br />That's another thing I'd like to learn: stating such problems. How to make sure that there is enough information in the problem that it could be solved, but not too much so that it wasn't too easy?<br />Can anyone post their problems here? Does one need to be the author of the problem? Or any problem will do?<br /><br />_"The crucial thing was to use the many cyclic quads on show and realize that EG//FH, then the rest was easy."_<br />OK I'll give it a shot myself firs, and then I'll check out your solution if I get stuck.Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-23420079243722388252016-08-21T03:53:50.396-07:002016-08-21T03:53:50.396-07:00Few tips....
Read Mind Map on this site.
What we ...Few tips....<br />Read Mind Map on this site.<br /><br />What we see is the end result of the Problem often much refined.<br /><br />The process of course is mainly trial and error, at times using many sketches. And then suddenly it dawns....<br /><br />Try to infer whatever u can from the given data, some or all of these inferences may lead u to the answer. <br /><br />Many a time these problems can be deviously difficult or most obvious.<br /><br />Let me explain my thought process on Antonio's latest Problem (1250) which appeared today in Gogeometry.<br /><br />The crucial thing was to use the many cyclic quads on show and realize that EG//FH, then the rest was easy.Sumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-15039573386392289602016-08-20T22:54:23.345-07:002016-08-20T22:54:23.345-07:00How can OD bisect <AOB? It isn't even conta...How can OD bisect <AOB? It isn't even contained in that angle :\Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-1058646786869576022016-08-20T14:41:55.204-07:002016-08-20T14:41:55.204-07:00Problem 1146 answer
Is triangleAEB=triangleABC (AB...Problem 1146 answer<br />Is triangleAEB=triangleABC (AB is common,AE=AC,<BAE=<BAC)<br />AE=EC (triangle AEC =equilateral ). ΑD=DC(<DAC=<DCA)<br /> ED perpendicular AC (AE=EC,AD=DC perpendicular bisector )<br /><ΒΕD=<AED-<AEB=30-20=10<br />BDCE is concyclic (<BED=10=<BCD)<br />APOSTOLIS MANOLOUDIS 4 HIGH SHCOOL OF KORYDALLOS PIRAEUS GREECE<br />APOSTOLIS MANOLOUDIShttps://www.blogger.com/profile/15561495997090211148noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-86030687269201214682016-08-20T01:11:44.651-07:002016-08-20T01:11:44.651-07:00Can you guys elaborate some more about the process...Can you guys elaborate some more about the process you went through to figure out that these particular auxiliary constructions can help to find the solution?<br />It's bad that people omit this important detail so often in their proofs, because then it is hard to learn from these solutions.<br />Or perhaps you can recommend some resources from which I could learn these techniques myself?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-4759191291589367932016-08-20T01:05:05.838-07:002016-08-20T01:05:05.838-07:00How did you know that CE=AF? What it follows from?...How did you know that CE=AF? What it follows from?Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-2939030155025497552016-04-11T13:32:07.218-07:002016-04-11T13:32:07.218-07:00.Forming equilateral triangle ACE (B is interior ....Forming equilateral triangle ACE (B is interior point triangle ACE).Then tri AEB= tri ABC, so <AEB=<ACB=20.But AE=EC, AD=DC then ED is perpendicular AC and <DEC=30. Therefore<br /><BED=<BCD=10 . So BDCE is concyclic. Then <DBC=<DEC=30.<br />APOSTOLIS MANOLOUDIShttps://www.blogger.com/profile/15561495997090211148noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-46616479791533461532015-11-06T12:14:26.207-08:002015-11-06T12:14:26.207-08:00Another solution: Extend (AB to P so that BP=PC; &...Another solution: Extend (AB to P so that BP=PC; <APC=80 degs, so D is the circumcenter of tr. ACP and subsequently tr. CPD is equilateral, making BP=PD, <BPD=20, thus <CBD=<PBD-<PBC=30 degs.<br /><br />Best regardsStan Fulgerhttps://www.facebook.com/stan.fulgernoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-26705756331144967602015-09-18T02:12:18.587-07:002015-09-18T02:12:18.587-07:00Further
5) if DN meets AE at P then P lies on ci...Further <br /><br />5) if DN meets AE at P then P lies on circle AMC whose centre is D<br /><br />6) if BD meets CE at Q then Tr. BQE is isoceles and BCQM is cyclic with diameter BQSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-59769213173280882132015-09-17T18:52:55.440-07:002015-09-17T18:52:55.440-07:00Thanx Ajith - sorry I saw your comments only just ...Thanx Ajith - sorry I saw your comments only just now. Sumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-35494740658933719602015-09-03T09:08:26.831-07:002015-09-03T09:08:26.831-07:00Excellent observations, Sumith!Excellent observations, Sumith!Ajithttps://www.blogger.com/profile/00611759721780927573noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-19859200103970282152015-08-25T22:20:09.781-07:002015-08-25T22:20:09.781-07:00We can also show that
1)BE is perpendicular to A...We can also show that <br /><br />1)BE is perpendicular to AD <br /><br />2)and if N is the point of intersection of AC and BE that DMNC is cyclic <br /><br />3)and that DN is perpendicular to AM<br /><br />4) and that D is the I centre of Tr. BNCSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-22765413719368524952015-08-25T21:53:22.641-07:002015-08-25T21:53:22.641-07:00Find E such that EA and EC are perpendicular to AB...Find E such that EA and EC are perpendicular to AB and BC respectively. Then ABCE is cyclic with M the mid point of diameter BE the centre of this circle. <br /><br />Then by computing the angles we can easily show that DM=DC from which we show that Tr.s MBD and CBD are congruent SSS and so BD bisects < MBC.<br />Hence x=30<br /><br />Sumith Peiris<br />Moratuwa<br />Sri LankaSumith Peirishttps://www.blogger.com/profile/06211995240466447227noreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-29844378407703980192015-08-21T00:15:41.173-07:002015-08-21T00:15:41.173-07:00Circumcircle ABC with center O, connect OA, OB, OC...Circumcircle ABC with center O, connect OA, OB, OC,OD;<br />From symmetry, OD bisect <AOB, so <AOD=50 degree;<br /><OAD=50 degree; so AD=DO;<br />Also AD=DC, so DO=DC;<br />Also tr OBC is equilateral tri;<br />From symmetry, BD bisect <OBC, X=30 degree<br />Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-55509998940487280142015-08-19T00:13:31.801-07:002015-08-19T00:13:31.801-07:00Oops, that's another problem!Oops, that's another problem!Stan Fulgerhttps://www.facebook.com/stan.fulgernoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-15325604083767254802015-08-18T13:16:43.369-07:002015-08-18T13:16:43.369-07:00In fact this is problem #2 from this site!In fact this is problem #2 from this site!Stan Fulgerhttps://www.facebook.com/stan.fulgernoreply@blogger.comtag:blogger.com,1999:blog-6933544261975483399.post-64356330426192044542015-08-15T05:04:18.569-07:002015-08-15T05:04:18.569-07:00Let AD intersect BC at E; <DEB=30 ( from tr. AC...Let AD intersect BC at E; <DEB=30 ( from tr. ACE) and let F be the reflection of A in CD; tr ACF is a 80-20-80 triangle and, well known, CE=AF and tr. ABF is isosceles giving BF=AF, hence BF=CE. But easily D is the circumcenter of tr ACF, so D lies onto perpendicular bisector of CF (and BE), tr. BDE is isosceles, giving <DBE=<DEB=30, done.Stan Fulgerhttps://www.facebook.com/stan.fulgernoreply@blogger.com