Do two new special points in any triangle exist?

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
2
down vote

favorite












There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.





Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by geogebra sofware. I proposed a conjecture:



In any triangle exist two points $P$ so that: $AA'=BB'=CC'$.



My question: Is the conjecture above correct?





enter image description here



My geogebra:



The Red locus: If $P$ lie on red locus then $AA'=CC'$.



The Blue locus: If $P$ lie on red locus then $AA'=BB'$.



The Pink locus: If $P$ lie on pink locus then $CC'=BB'$



See also:



  • Triangle centers





mg.metric-geometry geometry euclidean-geometry plane-geometry







share|cite|improve this question



















  • 1




    For equilateral triangle the number of such points is not 2 for sure
    – Fedor Petrov
    34 mins ago














up vote
2
down vote

favorite












There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.





Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by geogebra sofware. I proposed a conjecture:



In any triangle exist two points $P$ so that: $AA'=BB'=CC'$.



My question: Is the conjecture above correct?





enter image description here



My geogebra:



The Red locus: If $P$ lie on red locus then $AA'=CC'$.



The Blue locus: If $P$ lie on red locus then $AA'=BB'$.



The Pink locus: If $P$ lie on pink locus then $CC'=BB'$



See also:



  • Triangle centers





mg.metric-geometry geometry euclidean-geometry plane-geometry







share|cite|improve this question



















  • 1




    For equilateral triangle the number of such points is not 2 for sure
    – Fedor Petrov
    34 mins ago












up vote
2
down vote

favorite









up vote
2
down vote

favorite











There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.





Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by geogebra sofware. I proposed a conjecture:



In any triangle exist two points $P$ so that: $AA'=BB'=CC'$.



My question: Is the conjecture above correct?





enter image description here



My geogebra:



The Red locus: If $P$ lie on red locus then $AA'=CC'$.



The Blue locus: If $P$ lie on red locus then $AA'=BB'$.



The Pink locus: If $P$ lie on pink locus then $CC'=BB'$



See also:



  • Triangle centers





mg.metric-geometry geometry euclidean-geometry plane-geometry







share|cite|improve this question















There are some special points in any triangle, as Fermat point, symmedian point, incenter, Morley center, et cetera.





Let $P$ be a point on the plane, $PA$, $PB$, $PC$ meet $BC$, $CA$, $AB$ at $A'$, $B'$, $C'$ respectively. From my construction by geogebra sofware. I proposed a conjecture:



In any triangle exist two points $P$ so that: $AA'=BB'=CC'$.



My question: Is the conjecture above correct?





enter image description here



My geogebra:



The Red locus: If $P$ lie on red locus then $AA'=CC'$.



The Blue locus: If $P$ lie on red locus then $AA'=BB'$.



The Pink locus: If $P$ lie on pink locus then $CC'=BB'$



See also:



  • Triangle centers





mg.metric-geometry geometry euclidean-geometry plane-geometry




mg.metric-geometry geometry euclidean-geometry plane-geometry






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 36 mins ago









Fedor Petrov

45k5108213




45k5108213










asked 57 mins ago









Đào Thanh Oai

54218




54218







  • 1




    For equilateral triangle the number of such points is not 2 for sure
    – Fedor Petrov
    34 mins ago












  • 1




    For equilateral triangle the number of such points is not 2 for sure
    – Fedor Petrov
    34 mins ago







1




1




For equilateral triangle the number of such points is not 2 for sure
– Fedor Petrov
34 mins ago




For equilateral triangle the number of such points is not 2 for sure
– Fedor Petrov
34 mins ago










1 Answer
1






active

oldest

votes

















up vote
3
down vote



accepted










Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$






share|cite|improve this answer




















  • Thank You very much, this is new point. I have never seen the point before: researchgate.net/publication/…
    – Äào Thanh Oai
    2 mins ago










  • Oh no, This is old points anubih.ba/Journals/vol.11,no-1,y15/08Rev-Volenec-Kolar.pdf
    – Äào Thanh Oai
    just now










Your Answer




StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "504"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);













 

draft saved


draft discarded


















StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f311741%2fdo-two-new-special-points-in-any-triangle-exist%23new-answer', 'question_page');

);

Post as a guest

















1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes








up vote
3
down vote



accepted










Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$






share|cite|improve this answer




















  • Thank You very much, this is new point. I have never seen the point before: researchgate.net/publication/…
    – Äào Thanh Oai
    2 mins ago










  • Oh no, This is old points anubih.ba/Journals/vol.11,no-1,y15/08Rev-Volenec-Kolar.pdf
    – Äào Thanh Oai
    just now














up vote
3
down vote



accepted










Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$






share|cite|improve this answer




















  • Thank You very much, this is new point. I have never seen the point before: researchgate.net/publication/…
    – Äào Thanh Oai
    2 mins ago










  • Oh no, This is old points anubih.ba/Journals/vol.11,no-1,y15/08Rev-Volenec-Kolar.pdf
    – Äào Thanh Oai
    just now












up vote
3
down vote



accepted







up vote
3
down vote



accepted






Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$






share|cite|improve this answer












Your conjecture is true: these two points are sometimes called the equicevian points of $ABC.$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 33 mins ago









Donatien Bénéat

736




736











  • Thank You very much, this is new point. I have never seen the point before: researchgate.net/publication/…
    – Äào Thanh Oai
    2 mins ago










  • Oh no, This is old points anubih.ba/Journals/vol.11,no-1,y15/08Rev-Volenec-Kolar.pdf
    – Äào Thanh Oai
    just now
















  • Thank You very much, this is new point. I have never seen the point before: researchgate.net/publication/…
    – Äào Thanh Oai
    2 mins ago










  • Oh no, This is old points anubih.ba/Journals/vol.11,no-1,y15/08Rev-Volenec-Kolar.pdf
    – Äào Thanh Oai
    just now















Thank You very much, this is new point. I have never seen the point before: researchgate.net/publication/…
– Äào Thanh Oai
2 mins ago




Thank You very much, this is new point. I have never seen the point before: researchgate.net/publication/…
– Äào Thanh Oai
2 mins ago












Oh no, This is old points anubih.ba/Journals/vol.11,no-1,y15/08Rev-Volenec-Kolar.pdf
– Äào Thanh Oai
just now




Oh no, This is old points anubih.ba/Journals/vol.11,no-1,y15/08Rev-Volenec-Kolar.pdf
– Äào Thanh Oai
just now

















 

draft saved


draft discarded















































 


draft saved


draft discarded














StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f311741%2fdo-two-new-special-points-in-any-triangle-exist%23new-answer', 'question_page');

);

Post as a guest
































































Comments

Post a Comment

Popular posts from this blog

Long meetings (6-7 hours a day): Being “babysat” by supervisor

Image
Clash Royale CLAN TAG #URR8PPP .everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty margin-bottom:0; up vote 31 down vote favorite 4 I hold a PhD in mechanical engineering with a 7 year background in self-guided, self-paced research including 2 years as a graduate level course instructor and a subject matter expert for various engineering projects in a university setting in the United States. For the last 10-11 months, I am working in a research industry in France where my task is to debug C++ spaghetti code written over a period of several years by my current supervisor. It is a team of 2: just I and my supervisor. Diurnally, I am posed with a "bug of the day" to get out of this C++ code. My supervisor generally gives me about 15-30 minutes to solve it and then accosts me about whether or not the task is completed and then rinses and repeats this until the end of the day. Off-late, my supervisor has been holding 6-7 hour long "meetin
Read more

Is the Concept of Multiple Fantasy Races Scientifically Flawed? [closed]

Image
Clash Royale CLAN TAG #URR8PPP up vote 2 down vote favorite Many fantasy worlds have multiple humanoid species (elves, orcs, humans, etc) living in the same world, in addition to that, they often have the ability to interbreed. And I can't help but question their believability... fantasy-races reproduction interspecies-relations share | improve this question edited Aug 9 at 14:46 Michael Kjörling ♦ 21.1k 10 85 161 asked Aug 9 at 13:13 Chlodio 118 6 closed as unclear what you're asking by MichaelK, Gryphon, John, Vincent, Frostfyre Aug 9 at 15:25 Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question. 3 Stack Exchange exist to
Read more

Confectionery

Image
Prepared foods made with a lot of sugar or other cabohydrates "Sweetmeat" redirects here. For the racehorse, see Sweetmeat (horse). Not to be confused with Sweetbread. This Kransekake is a traditional Scandinavian baker's confection. Confectionery is the art of making confections , which are food items that are rich in sugar and carbohydrates. Exact definitions are difficult. [1] In general, though, confectionery is divided into two broad and somewhat overlapping categories, bakers' confections and sugar confections. [2] Bakers' confectionery, also called flour confections , includes principally sweet pastries, cakes, and similar baked goods. Sugar confectionery includes candies ( sweets in British English), candied nuts, chocolates, chewing gum, bubble gum, pastillage, and other confections that are made primarily of sugar. In some cases, chocolate confections (confections made of chocolate) are treated as a separate category, as are sugar-free versions of
Read more