Do two new special points in any triangle exist?

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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













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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

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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










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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

















 

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