Heartsuited Trajectory: Gergonne Triangle of the Isosceles Triangle
by Klaus-Anton, Jan 20, 2022, 10:25 AM
Inspiration: Ph. Ivaldi here: anim.asy,
(Please note [now in the web.archive.org!!]: http://www.piprime.fr/asymptote/).
The triangle module of geometry.asy defines a triangle which inside of this module there gets the name intouch. I have found out that this triangle also is called Gergonne Triangle (Mathematik alphaLexikon and there in 07 Dreieck Theorie, Sätze und Grundlagen, Rechtwinkliges Dreieck, Dreieckskonstruktion (146 Seiten, PDF). - 07-Dreieck.pdf.
There on logical page 24 (which corresponends to page 771 of the encyclopedia) on the bottom it is said: „Das Dreieck
, das durch die Berührungspunkte des Inkreises gebildet wird, heißt Gergonne-Dreieck.“
In the case that the following animiation does not run for you, reopen this blog entry in a new tab of your browser or make a right mouseclick on the figure to open it on imgur. I drew here the Gergonne Triangle in green.
The figure is outgoing from the isosceles triangle. So because of this restriction one perhaps in fact will not get all Gergonne Triangles. You see. here i am actually not sure. If i am not right, the word general Gergonne Triangle is too wide, or true inspite of this in an only relative sense. If you see more generality, please let me know it.
Please: Be aware that in the case when M=N=P there is no triangle and so there also cannot be any incircle. As logical consequence the heartsuited curve will not be defined at the origin and the east pol of the big circle. In the somewhat alternative sense you count the east pol twice - as the begin point and as the end point - this together with the origin as the midpoint, there are in total three undefined points.
Having this restriction in mind: The Point P walks from the east pol (1,0) to the center of the big circle (0,0) and from there then again back to the east pol.
The point N has as trajectory the circle ( (0,0), (1,0 ) ). Marking the point M at every position makes you out the heartsuited trajectory.
The incircle is drawn in blue. The nearer it is to the east or west pol the smaller it is untill it disappears. It reappears when these pols are walked through. So long as this circle goes from right to left it is in the upper semicicle, and when it goes from left to right this happens in the lower semicircle.
(Note 1: Phillipe Ivaldi did not draw the circled trajectory of the point N. In gray here i have marked me the radii at -54°, -60°, and -120°.)
(Note 2: Phillipe Ivaldi discuussed this on AoPS. Go to this link: Drawing loci.)
Here you see the unanimated version of Phillipe Ivaldi he presented in the link of above. This looks somewhat confusing. Showing the same thing with three different situations.
![[asy]
/*https://artofproblemsolving.com/community/c68h176847p1012294
modified by AoPS user Klaus-Anton */
import geometry;
size(6cm,0);
circle C=circle(origin,1);
draw(C);
point A=point(C,0), B, M;
guide locus;
real eps=0.001;
for (real a=eps; a < 360; a += 1) {
B=angpoint(C,a);
triangle t=triangle(origin,A,B);
triangle intouch=intouch(t);
M=intouch(t.AC);
locus=locus..M;
if(a-eps == 30 || a-eps == 90 || a-eps == 225 || a-eps == 345) {
draw(t, dot);
draw(intouch, dot);
draw(incircle(t), 0.8*blue);
label("$N$", Label("$M$", 0.8*red), "$P$", intouch);
dot(M, 0.8*red);
}
}
draw(locus..cycle, bp+0.8*red);
[/asy]](//latex.artofproblemsolving.com/3/4/8/3484d67b9294a1eeca55ae4612518f747762d503.png)
Consider the contained code snippet before here is drawn and labelled:
As all this much motion makes iritatiom here is a static image of the incircle in the Domino-Triangle (lengths: 1, 2, sqrt(5)):
![[asy]
import geometry;
size(6cm,0);
triangle t=triangleabc(1,2,sqrt(5));
circle C=circumcircle(t), inC=incircle(t);
draw(inC, bp+0.8*red);
draw(C, bp+0.8*blue);
draw(t, linewidth(bp));
draw(inC*t, linewidth(bp));
//draw(inC*C, bp+0.8*blue+1mm);
label(t);
shipout(bbox(.23cm, Fill(white)));
[/asy]](//latex.artofproblemsolving.com/7/b/5/7b5ca30635c8a0b5ebea346debfe95f2a71e8630.png)
![[asy]
import geometry;
size(6cm,0);
triangle t=triangleabc(1,2,sqrt(5));
circle C=circumcircle(t), inC=incircle(t);
draw(inC, bp+0.8*red);
draw(C, bp+0.8*blue);
draw(t, linewidth(bp));
draw(inC*t, linewidth(bp));
draw(inC*C, bp+0.8*blue);
label(t);
shipout(bbox(.23cm, Fill(white)));
[/asy]](//latex.artofproblemsolving.com/3/d/8/3d8b68c9bcae9d6eb63943ed57fd78b7a607fbf7.png)
![[asy]
import geometry;
size(6cm,0);
triangle t=triangleabc(1,2,sqrt(5));
circle C=circumcircle(t), inC=incircle(t);
draw(inC, bp+0.8*red);
dot(C.C, bp+0.8*blue);
draw(t.A--bisectorpoint(t.CB),red);
draw(t.B--bisectorpoint(t.CA),red);
draw(t.C--bisectorpoint(t.AB),red);
draw(t, linewidth(bp));
draw(inC*t, linewidth(bp));
draw(inC*C.C, bp+0.8*blue+1mm);
//draw(inC*C, bp+0.8*red+1mm);
dot(incenter(t));
label(t);
shipout(bbox(.23cm, Fill(white)));
[/asy]](//latex.artofproblemsolving.com/f/a/f/faf56574bd183972fd268c5400d586f0e85ff30a.png)
Further reading is here: Further Steps towards the Gergonne-Triangle in the Isosceles Triangle.
Edit-2023-17-01 / (first comment Mar 6, 2022):
Here is the link to the imgur-mp4-video. There you can make it play faster or slower ..., use right mouse click for options.
Edit-2023-17-01 / (second comment Mar 6, 2022): These ideas actually have been picked up and further elaborated by Hans Walser, Herzkurve6, (Walser 2022). And also in Achterschleife (Walser 2022).
(Please note [now in the web.archive.org!!]: http://www.piprime.fr/asymptote/).
The triangle module of geometry.asy defines a triangle which inside of this module there gets the name intouch. I have found out that this triangle also is called Gergonne Triangle (Mathematik alphaLexikon and there in 07 Dreieck Theorie, Sätze und Grundlagen, Rechtwinkliges Dreieck, Dreieckskonstruktion (146 Seiten, PDF). - 07-Dreieck.pdf.
There on logical page 24 (which corresponends to page 771 of the encyclopedia) on the bottom it is said: „Das Dreieck
, das durch die Berührungspunkte des Inkreises gebildet wird, heißt Gergonne-Dreieck.“In the case that the following animiation does not run for you, reopen this blog entry in a new tab of your browser or make a right mouseclick on the figure to open it on imgur. I drew here the Gergonne Triangle in green.
The figure is outgoing from the isosceles triangle. So because of this restriction one perhaps in fact will not get all Gergonne Triangles. You see. here i am actually not sure. If i am not right, the word general Gergonne Triangle is too wide, or true inspite of this in an only relative sense. If you see more generality, please let me know it.
Please: Be aware that in the case when M=N=P there is no triangle and so there also cannot be any incircle. As logical consequence the heartsuited curve will not be defined at the origin and the east pol of the big circle. In the somewhat alternative sense you count the east pol twice - as the begin point and as the end point - this together with the origin as the midpoint, there are in total three undefined points.
Having this restriction in mind: The Point P walks from the east pol (1,0) to the center of the big circle (0,0) and from there then again back to the east pol.
The point N has as trajectory the circle ( (0,0), (1,0 ) ). Marking the point M at every position makes you out the heartsuited trajectory.
The incircle is drawn in blue. The nearer it is to the east or west pol the smaller it is untill it disappears. It reappears when these pols are walked through. So long as this circle goes from right to left it is in the upper semicicle, and when it goes from left to right this happens in the lower semicircle.
(Note 1: Phillipe Ivaldi did not draw the circled trajectory of the point N. In gray here i have marked me the radii at -54°, -60°, and -120°.)
(Note 2: Phillipe Ivaldi discuussed this on AoPS. Go to this link: Drawing loci.)
Here you see the unanimated version of Phillipe Ivaldi he presented in the link of above. This looks somewhat confusing. Showing the same thing with three different situations.
![[asy]
/*https://artofproblemsolving.com/community/c68h176847p1012294
modified by AoPS user Klaus-Anton */
import geometry;
size(6cm,0);
circle C=circle(origin,1);
draw(C);
point A=point(C,0), B, M;
guide locus;
real eps=0.001;
for (real a=eps; a < 360; a += 1) {
B=angpoint(C,a);
triangle t=triangle(origin,A,B);
triangle intouch=intouch(t);
M=intouch(t.AC);
locus=locus..M;
if(a-eps == 30 || a-eps == 90 || a-eps == 225 || a-eps == 345) {
draw(t, dot);
draw(intouch, dot);
draw(incircle(t), 0.8*blue);
label("$N$", Label("$M$", 0.8*red), "$P$", intouch);
dot(M, 0.8*red);
}
}
draw(locus..cycle, bp+0.8*red);
[/asy]](http://latex.artofproblemsolving.com/3/4/8/3484d67b9294a1eeca55ae4612518f747762d503.png)
Consider the contained code snippet before here is drawn and labelled:
if(a-eps == 30 || a-eps == 90 || a-eps == 225 || a-eps == 345) { [...] }
As all this much motion makes iritatiom here is a static image of the incircle in the Domino-Triangle (lengths: 1, 2, sqrt(5)):
![[asy]
import geometry;
size(6cm,0);
triangle t=triangleabc(1,2,sqrt(5));
circle C=circumcircle(t), inC=incircle(t);
draw(inC, bp+0.8*red);
draw(C, bp+0.8*blue);
draw(t, linewidth(bp));
draw(inC*t, linewidth(bp));
//draw(inC*C, bp+0.8*blue+1mm);
label(t);
shipout(bbox(.23cm, Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/7/b/5/7b5ca30635c8a0b5ebea346debfe95f2a71e8630.png)
![[asy]
import geometry;
size(6cm,0);
triangle t=triangleabc(1,2,sqrt(5));
circle C=circumcircle(t), inC=incircle(t);
draw(inC, bp+0.8*red);
draw(C, bp+0.8*blue);
draw(t, linewidth(bp));
draw(inC*t, linewidth(bp));
draw(inC*C, bp+0.8*blue);
label(t);
shipout(bbox(.23cm, Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/3/d/8/3d8b68c9bcae9d6eb63943ed57fd78b7a607fbf7.png)
![[asy]
import geometry;
size(6cm,0);
triangle t=triangleabc(1,2,sqrt(5));
circle C=circumcircle(t), inC=incircle(t);
draw(inC, bp+0.8*red);
dot(C.C, bp+0.8*blue);
draw(t.A--bisectorpoint(t.CB),red);
draw(t.B--bisectorpoint(t.CA),red);
draw(t.C--bisectorpoint(t.AB),red);
draw(t, linewidth(bp));
draw(inC*t, linewidth(bp));
draw(inC*C.C, bp+0.8*blue+1mm);
//draw(inC*C, bp+0.8*red+1mm);
dot(incenter(t));
label(t);
shipout(bbox(.23cm, Fill(white)));
[/asy]](http://latex.artofproblemsolving.com/f/a/f/faf56574bd183972fd268c5400d586f0e85ff30a.png)
Further reading is here: Further Steps towards the Gergonne-Triangle in the Isosceles Triangle.
Edit-2023-17-01 / (first comment Mar 6, 2022):
Here is the link to the imgur-mp4-video. There you can make it play faster or slower ..., use right mouse click for options.
Edit-2023-17-01 / (second comment Mar 6, 2022): These ideas actually have been picked up and further elaborated by Hans Walser, Herzkurve6, (Walser 2022). And also in Achterschleife (Walser 2022).
This post has been edited 37 times. Last edited by Klaus-Anton, Jan 17, 2023, 9:34 AM
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