The Loci of the First and Second Fermat Point in the Isosceles Triangle
by Klaus-Anton, Jun 16, 2024, 2:04 PM
You can draw with asy a locus of a moving point. I have made use of this in Heartsuited Trajectory: Gergonne Triangle of the Isosceles Triangle.
[On AoPS look too into the link Drawing loci, and there in especially the answer of Ph. Ivaldi.]
Here in Figure 2 for some reason the asy path connector ".." seems to make confusion.
What do we see?
On the right hand there are drawn two equilateral triangles with sidelength
. The base of these triangles is the segment from the origin with
to the point
.
Partially in red or blue there appear the loci of the First and the Second Fermat Point as the circumcircles of the triangles with circumcircle centers at
. The circumcircle has as radius
.
The circumcircle of the upper indrawn triangle goes with his southpol through the midpoint of the lower triangle. And correspondending to this - the circumcircle of the lower indrawn triangle goes with his northpol through the midpoint of the upper triangle.
The blue trajectory of the First Fermat Point forms a fish - divided into body and tailfin. It is symmetrical to the xaxis. The conjugated parts are made up out of semicircles.
And on the left hand? - What do we see there?
What is it? That what there comes to appearence? - A something.
Sidenote: The philosopher Immanel Kant says in his Kritik der reinen Vernunft:
(Kant KpV 1781 / 1787
Der transzendentalen Elementarlehre erster Teil
Die transzendentale Ästhetik
§ 1, (book typographically the second paragraph of the logical § 1 of the table of contents))
"The effect of an object on the ability to imagine, to the extent that we are affected by it, is sensation. The intuition which relates to the object through sensation is called empirical. The indeterminate object of an empirical view is called appearance." (Google translate)
So. What is there, what we can see?
We see white and gray shaded in a special sense rouned lines. More or less thick. These lines seem me to give different approximations of the in that corner coming together of:
The way of these rounded curves remember somewhat Pascal Snakes and the classical Cardioid. But in my eyes i find them more edged. Also you could say - if you want so - more "modern".
The whiter gray shadings seem me to begin the point
. Nearer and nearer to the point
- the west pol of our great circle - there is the very most white gray shading. It has the form of a rounded funnel. Okay. For now so far. We let it here actually. Without really beeing able to explain it.
Note:
The two here new appearing circles remember me Study arcsubtended.
[On AoPS look too into the link Drawing loci, and there in especially the answer of Ph. Ivaldi.]
![[asy]
size(6cm,0);
import geometry;
circle C=circle(origin,1);
draw(C,linewidth(3));
// A=(1,0)
point A=point(C,0), B, F1;
point FirstFermatPoint;
guide locus;
for (real a=0.001; a < 360; a += 12) {
//save();
B=angpoint(C,a);
triangle t=triangle(origin,A,B);
draw(t,gray);
// dot(fermat(t),blue);//works
dot(fermat(t)[0],blue);
dot(fermat(t)[1],red);
//dot(fermat(t));
//point F1=fermat(t);
FirstFermatPoint=fermat(t)[0];
locus=locus..FirstFermatPoint;
// draw(locus, bp+0.8*blue);
}
draw(origin--dir(30)
^^dir(60)--(dir(60).x,0)
);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)-1)/2;
draw(circle(A-1,.5*sqrt(3))
^^circle(A-1,sqrt(3)/3)
^^S--N
,linewidth(gr^.5));
draw(arc(origin,.5*sqrt(3),0,120)
^^arc(origin,sqrt(3)/3,0,120)
,white+linewidth(gr^.5));
draw((1,0)--origin--dir(60)--(1,0)--dir(-60)--origin,linewidth(1.5)+red);
dotfactor=12;
dot(origin
^^A^^dir(60)^^dir(-60)
,red+linewidth(gr^.5), Fill(white));
shipout(bbox(2mm,FillDraw(white,white)));
[/asy]](http://latex.artofproblemsolving.com/e/4/a/e4a7146ef8634921e8a68bb4689474e9473d70ff.png)
Figure 1
![[asy]
size(6cm,0);
import geometry;
circle C=circle(origin,1);
draw(C,linewidth(3));
// A=(1,0)
point A=point(C,0), B, F1;
point FirstFermatPoint;
guide locus;
for (real a=0.001; a < 360; a += 1) {
//save();
B=angpoint(C,a);
triangle t=triangle(origin,A,B);
draw(t,gray);
// dot(fermat(t),blue);//works
dot(fermat(t)[0],blue);
dot(fermat(t)[1],red);
//dot(fermat(t));
//point F1=fermat(t);
FirstFermatPoint=fermat(t)[0];
locus=locus..FirstFermatPoint;
draw(locus, bp+0.8*blue);
}
draw(origin--dir(30)
^^dir(60)--(dir(60).x,0)
);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)-1)/2;
draw(circle(A-1,.5*sqrt(3))
^^circle(A-1,sqrt(3)/3)
^^S--N
,linewidth(gr^.5));
draw(arc(origin,.5*sqrt(3),0,120)
^^arc(origin,sqrt(3)/3,0,120)
,white+linewidth(gr^.5));
draw((1,0)--origin--dir(60)--(1,0)--dir(-60)--origin,linewidth(1.5)+red);
dotfactor=12;
dot(origin
^^A^^dir(60)^^dir(-60)
,red+linewidth(gr^.5), Fill(white));
shipout(bbox(2mm,FillDraw(white,white)));
[/asy]](http://latex.artofproblemsolving.com/5/6/2/562522c45b69f469f6aff654fa2246315dab0a79.png)
Figure 2
Here in Figure 2 for some reason the asy path connector ".." seems to make confusion.
![[asy]
size(6cm,0);
import geometry;
circle C=circle(origin,1);
draw(C,linewidth(3));
// A=(1,0)
point A=point(C,0), B, F1;
point FirstFermatPoint;
guide locus;
guide locus1;
guide locus2;
for (real a=0.001; a < 360; a += 1) {
//save();
B=angpoint(C,a);
triangle t=triangle(origin,A,B);
draw(t,gray);
// dot(fermat(t),blue);//works
picture pic;
dot(pic,fermat(t)[0],blue);
dot(pic,fermat(t)[1],red);
//add(pic);
//dot(fermat(t));
//point F1=fermat(t);
FirstFermatPoint=fermat(t)[0];
point SecondFermatPoint=fermat(t)[1];
locus1=locus1--FirstFermatPoint;
locus2=locus2--SecondFermatPoint;
draw(locus1, bp+0.8*blue);
draw(locus2, bp+0.8*red);
}
draw(origin--dir(30)
^^dir(60)--(dir(60).x,0)
);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)-1)/2;
draw(circle(A-1,.5*sqrt(3))
^^circle(A-1,sqrt(3)/3)
^^S--N
,linewidth(gr^.5));
draw(arc(origin,.5*sqrt(3),0,120)
^^arc(origin,sqrt(3)/3,0,120)
,white+linewidth(gr^.5));
draw((1,0)--origin--dir(60)--(1,0)--dir(-60)--origin,linewidth(1.5)+red);
dotfactor=12;
dot(origin
^^A^^dir(60)^^dir(-60)
,red+linewidth(gr^.5), Fill(white));
dot((1,1),invisible);// Only here for correct rendering on AoPS
//add(pic);
dot((sqrt(3)/3,0));
dot((.5,sqrt(3)/6));
usepackage("kpfonts");
label("$\sqrt3/3$",(sqrt(3)/5.25,0),.4S,red);
dot((0,sqrt(3)/3));
dot((0,-sqrt(3)/3));
dot((.5,-sqrt(3)/6));
draw((sqrt(3)/3,0)
^^(.5,sqrt(3)/6)
^^(0,sqrt(3)/3)
^^(0,-sqrt(3)/3)
^^(.5,-sqrt(3)/6)
,white+linewidth(gr^.5));
dot((0,sqrt(3)/3)//absolutes Maximum blauer Fisch
^^(0,-sqrt(3)/3)//absolutes Minimum blauer Fisch
^^(.5,sqrt(3)/6)//relatives Maximum blauer Fisch
^^(.5,-sqrt(3)/6)//relatives Minimum blauer Fisch
,blue);
draw((0,sqrt(3)/3)//absolutes Maximum blauer Fisch
^^(0,-sqrt(3)/3)//absolutes Minimum blauer Fisch
^^(.5,sqrt(3)/6)//relatives Maximum blauer Fisch
^^(.5,-sqrt(3)/6)//relatives Minimum blauer Fisch
,linewidth(gr^.5)+white);
dot(( -(sqrt(3)/3-.5),sqrt(3)/6 ),blue);
draw(( -(sqrt(3)/3-.5),sqrt(3)/6 ),linewidth(gr^.5)+white);
//Linker oberer Extempunkt blauer Fisch
//(sqrt(3)/3)-.5= 0.0773502691896258...
dot(( -(sqrt(3)/3-.5),-sqrt(3)/6 ),blue);
draw(( -(sqrt(3)/3-.5),-sqrt(3)/6 ),linewidth(gr^.5)+white);
//Linker unterer Extempunkt blauer Fisch
//(sqrt(3)/3)-.5= 0.0773502691896258...
draw(origin^^(1,0)^^dir(60)^^dir(-60),linewidth(gr^.5));
shipout(bbox(2mm,FillDraw(white,white)));
[/asy]](http://latex.artofproblemsolving.com/4/b/c/4bca95573de88ceba1c107be0c65993db594a93f.png)
Figure 3
asy path connector "
"

Blue Partialcircles: Trajectory of First Fermat Point
Red Partialcircles: Trajectory of Second Fermat Point
What do we see?
On the right hand there are drawn two equilateral triangles with sidelength



Partially in red or blue there appear the loci of the First and the Second Fermat Point as the circumcircles of the triangles with circumcircle centers at


The circumcircle of the upper indrawn triangle goes with his southpol through the midpoint of the lower triangle. And correspondending to this - the circumcircle of the lower indrawn triangle goes with his northpol through the midpoint of the upper triangle.
The blue trajectory of the First Fermat Point forms a fish - divided into body and tailfin. It is symmetrical to the xaxis. The conjugated parts are made up out of semicircles.
![[asy]
size(6cm,0);
import geometry;
circle C=circle(origin,1);
draw(C,linewidth(3));
// A=(1,0)
point A=point(C,0), B, F1;
point FirstFermatPoint;
guide locus;
guide locus1;
guide locus2;
for (real a=0.001; a < 360; a += 1) {
//save();
B=angpoint(C,a);
triangle t=triangle(origin,A,B);
draw(t,gray);
// dot(fermat(t),blue);//works
picture pic;
dot(pic,fermat(t)[0],blue);
dot(pic,fermat(t)[1],red);
//add(pic);
//dot(fermat(t));
//point F1=fermat(t);
FirstFermatPoint=fermat(t)[0];
point SecondFermatPoint=fermat(t)[1];
locus1=locus1--FirstFermatPoint;
locus2=locus2--SecondFermatPoint;
draw(locus1, bp+0.8*blue);
draw(locus2, bp+0.8*red);
}
draw(origin--dir(30)
^^dir(60)--(dir(60).x,0)
);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)-1)/2;
draw(circle(A-1,.5*sqrt(3))
^^circle(A-1,sqrt(3)/3)
^^S--N
,linewidth(gr^.5));
draw(scale(sqrt(3)/3)*polygon(3)
^^rotate(60)*scale(sqrt(3)/3)*polygon(3),cyan);
draw(arc(origin,.5*sqrt(3),0,120)
^^arc(origin,sqrt(3)/3,0,120)
,white+linewidth(gr^.5));
draw((1,0)--origin--dir(60)--(1,0)--dir(-60)--origin,linewidth(1.5)+red);
dotfactor=12;
dot(origin
^^A^^dir(60)^^dir(-60)
,red+linewidth(gr^.5), Fill(white));
dot((1,1),invisible);// Only here for correct rendering on AoPS
//add(pic);
dot((sqrt(3)/3,0));
dot((.5,sqrt(3)/6));
usepackage("kpfonts");
label("$\sqrt3/3$",(sqrt(3)/5.25,0),.4S,red);
dot((0,sqrt(3)/3));
dot((0,-sqrt(3)/3));
dot((.5,-sqrt(3)/6));
draw((sqrt(3)/3,0)
^^(.5,sqrt(3)/6)
^^(0,sqrt(3)/3)
^^(0,-sqrt(3)/3)
^^(.5,-sqrt(3)/6)
,white+linewidth(gr^.5));
dot((0,sqrt(3)/3)//absolutes Maximum blauer Fisch
^^(0,-sqrt(3)/3)//absolutes Minimum blauer Fisch
^^(.5,sqrt(3)/6)//relatives Maximum blauer Fisch
^^(.5,-sqrt(3)/6)//relatives Minimum blauer Fisch
,blue);
draw((0,sqrt(3)/3)//absolutes Maximum blauer Fisch
^^(0,-sqrt(3)/3)//absolutes Minimum blauer Fisch
^^(.5,sqrt(3)/6)//relatives Maximum blauer Fisch
^^(.5,-sqrt(3)/6)//relatives Minimum blauer Fisch
,linewidth(gr^.5)+white);
dot(( -(sqrt(3)/3-.5),sqrt(3)/6 ),blue);
draw(( -(sqrt(3)/3-.5),sqrt(3)/6 ),linewidth(gr^.5)+white);
//Linker oberer Extempunkt blauer Fisch
//(sqrt(3)/3)-.5= 0.0773502691896258...
dot(( -(sqrt(3)/3-.5),-sqrt(3)/6 ),blue);
draw(( -(sqrt(3)/3-.5),-sqrt(3)/6 ),linewidth(gr^.5)+white);
//Linker unterer Extempunkt blauer Fisch
//(sqrt(3)/3)-.5= 0.0773502691896258...
draw(origin^^(1,0)^^dir(60)^^dir(-60),linewidth(gr^.5));
shipout(bbox(2mm,FillDraw(white,white)));
[/asy]](http://latex.artofproblemsolving.com/b/1/1/b118192bac0cd2ce87eaacb897ecd55cd9a31967.png)
Figure 4
Cyan Star added made up out of two by
about the origin rotated Triangles with the same Size as the red Triangles

![[asy]
size(6cm,0);
import geometry;
circle C=circle(origin,1);
draw(C,linewidth(3));
// A=(1,0)
point A=point(C,0), B, F1;
point FirstFermatPoint;
guide locus;
guide locus1;
guide locus2;
for (real a=0.001; a < 360; a += 1) {
//save();
B=angpoint(C,a);
triangle t=triangle(origin,A,B);
draw(t,gray);
// dot(fermat(t),blue);//works
picture pic;
dot(pic,fermat(t)[0],blue);
dot(pic,fermat(t)[1],red);
//add(pic);
//dot(fermat(t));
//point F1=fermat(t);
FirstFermatPoint=fermat(t)[0];
point SecondFermatPoint=fermat(t)[1];
locus1=locus1--FirstFermatPoint;
locus2=locus2--SecondFermatPoint;
draw(locus1, bp+0.8*blue);
draw(locus2, bp+0.8*red);
}
draw(origin--dir(30)
^^dir(60)--(dir(60).x,0)
);
real gr=(sqrt(5)-1)/2;
real GR=(sqrt(5)-1)/2;
draw(circle(A-1,.5*sqrt(3))
^^circle(A-1,sqrt(3)/3)
^^S--N
,linewidth(gr^.5));
draw(scale(sqrt(3)/3)*polygon(3)
^^rotate(60)*scale(sqrt(3)/3)*polygon(3),cyan);
draw(dir(60)--dir(120)--origin
--dir(240)--dir(-60)
,green+linewidth(1+gr^2));//Sigma!!!
draw(shift(sqrt(3)/3*dir(30))*scale(sqrt(3)/3)*polygon(3)
^^shift(sqrt(3)/3*dir(-30))*(rotate(60)*scale(sqrt(3)/3)*polygon(3)),cyan);
draw(arc(origin,.5*sqrt(3),0,120)
^^arc(origin,sqrt(3)/3,0,120)
,white+linewidth(gr^.5));
draw((1,0)--origin--dir(60)--(1,0)--dir(-60)--origin,linewidth(1.5)+red);
dotfactor=12;
dot(origin
^^A^^dir(60)^^dir(-60)
,red+linewidth(gr^.5), Fill(white));
dot((1,1),invisible);// Only here for correct rendering on AoPS
//add(pic);
dot((sqrt(3)/3,0));
dot((.5,sqrt(3)/6));
usepackage("kpfonts");
label("$\sqrt3/3$",(sqrt(3)/5.25,0),.4S,red);
dot((0,sqrt(3)/3));
dot((0,-sqrt(3)/3));
dot((.5,-sqrt(3)/6));
draw((sqrt(3)/3,0)
^^(.5,sqrt(3)/6)
^^(0,sqrt(3)/3)
^^(0,-sqrt(3)/3)
^^(.5,-sqrt(3)/6)
,white+linewidth(gr^.5));
dot((0,sqrt(3)/3)//absolutes Maximum blauer Fisch
^^(0,-sqrt(3)/3)//absolutes Minimum blauer Fisch
^^(.5,sqrt(3)/6)//relatives Maximum blauer Fisch
^^(.5,-sqrt(3)/6)//relatives Minimum blauer Fisch
,blue);
draw((0,sqrt(3)/3)//absolutes Maximum blauer Fisch
^^(0,-sqrt(3)/3)//absolutes Minimum blauer Fisch
^^(.5,sqrt(3)/6)//relatives Maximum blauer Fisch
^^(.5,-sqrt(3)/6)//relatives Minimum blauer Fisch
,linewidth(gr^.5)+white);
dot(( -(sqrt(3)/3-.5),sqrt(3)/6 ),blue);
draw(( -(sqrt(3)/3-.5),sqrt(3)/6 ),linewidth(gr^.5)+white);
//Linker oberer Extempunkt blauer Fisch
//(sqrt(3)/3)-.5= 0.0773502691896258...
dot(( -(sqrt(3)/3-.5),-sqrt(3)/6 ),blue);
draw(( -(sqrt(3)/3-.5),-sqrt(3)/6 ),linewidth(gr^.5)+white);
//Linker unterer Extempunkt blauer Fisch
//(sqrt(3)/3)-.5= 0.0773502691896258...
draw(origin^^(1,0)^^dir(60)^^dir(-60),linewidth(gr^.5));
/*draw(rotate(-60)*(shift(sqrt(3)/3*dir(30))*scale(sqrt(3)/3))*polygon(3)
^^rotate(-60)*(shift(sqrt(3)/3*dir(-30))*(rotate(60)*scale(sqrt(3)/3))*polygon(3)),cyan);
*/
/*
draw(dir(60)--dir(120)--origin
--dir(240)--dir(-60)
,green+linewidth(1+gr^2));//Sigma!!!
*/
//draw(parallel(1/20*dir(60),line(dir(120),dir(-60))));//evtl.
shipout(bbox(2mm,FillDraw(white,white)));
[/asy]](http://latex.artofproblemsolving.com/a/f/a/afac2eb040cf900a54234fc88c058a42d33d84e9.png)
Figure 5
The greek Letter Sigma (in green)
And on the left hand? - What do we see there?
What is it? That what there comes to appearence? - A something.
Sidenote: The philosopher Immanel Kant says in his Kritik der reinen Vernunft:
Kant wrote:
Die Wirkung eines Gegenstandes auf die Vorstellungsfähigkeit, so fern wir von demselben affiziert werden, ist Empfindung. Diejenige Anschauung, welche sich auf den Gegenstand durch Empfindung bezieht, heißt empirisch. Der unbestimmte Gegenstand einer empirischen Anschauung heißt Erscheinung.
(Kant KpV 1781 / 1787
Der transzendentalen Elementarlehre erster Teil
Die transzendentale Ästhetik
§ 1, (book typographically the second paragraph of the logical § 1 of the table of contents))
"The effect of an object on the ability to imagine, to the extent that we are affected by it, is sensation. The intuition which relates to the object through sensation is called empirical. The indeterminate object of an empirical view is called appearance." (Google translate)
So. What is there, what we can see?
We see white and gray shaded in a special sense rouned lines. More or less thick. These lines seem me to give different approximations of the in that corner coming together of:
- The arc centered at the origin with radius one and going from degrees 60° to degrees 180°
- The xaxis
The way of these rounded curves remember somewhat Pascal Snakes and the classical Cardioid. But in my eyes i find them more edged. Also you could say - if you want so - more "modern".
The whiter gray shadings seem me to begin the point


Note:
The two here new appearing circles remember me Study arcsubtended.
This post has been edited 25 times. Last edited by Klaus-Anton, Jun 22, 2024, 3:46 PM