Resumen

Source

The main reason for this diagram is to emphasise that the Fermat point is NOT the same as the first isogonic center which is (unfortunately) a very widespread misconception. The source is licensed under the same license as the image. Feel free to edit, fix, or improve it!

fermat_point_scope.euk

% Combination of 3 figures in 1. 

box(0,0,10,24)
draw(segment(point(0,9),point(10,9)))
draw(segment(point(0,16),point(10,16)))

% Proof that the Fermat point lies within the triangle
A = point(7,21)
B = point(1,18)
C = point(9,18)
D = point(4,17)
E = point(7.5,23)
thickness(2)
draw(line(B,C))
draw(line(A,B))
draw(line(A,C))
thickness(0.5)
draw(segment(D,A))
draw(segment(D,B))
draw(segment(D,C))
draw(segment(E,A))
draw(segment(E,B))
draw(segment(E,C))
label(A,0.2,170:)
label(B,0.2,110:)
label(C,0.2,45:)
draw($P'$,intersection(line(A,D),line(B,C)),0.1,280:)
draw($P$,D,0.1,270:)
draw("\textbf{A-zone only}",D,2,350:)
draw($P'$,A,0.2,355:)
draw($P$,E,0.2,0:)
draw("\textbf{B-zone and C-zone}",E,0.4,90:)
draw("\textbf{C-zone only}",segment(A,B),2,120:)

% Case 1  -  an angle at least 120 degrees
H = point(6,11.5)
I = point(1,10)
J = point(9,10)
K = point(4.8,10.5)
I H F equilateral
K H G equilateral
thickness(2)
draw(H,I,J)
thickness(0.5)
draw(H,K,G)
draw(segment(F,H))
draw(segment(F,I))
draw(segment(F,G))
draw(segment(K,I))
draw(segment(K,J))
draw($A$,H,0.2,20:)
draw($B$,I,0.2,180:)
draw($C$,J,0.2,0:)
draw($P$,K,0.1,270:)
draw($Q$,G,0.1,210:)
draw($F$,F,0.2,180:)
draw("\textbf{\Huge Case 1}",F,4.5,0:)
mark(segment(K,I),simple,0.7)
mark(segment(G,F),simple,0.7)
mark(segment(H,K),double,0.7)
mark(segment(K,G),double,0.7)
mark(segment(G,H),double,0.7)
mark(segment(H,I),triple,0.7)
mark(segment(I,F),triple,0.7)
mark(segment(F,H),triple,0.7)

% Case 2  -  no angle greater than 120 degrees
L = point(4,8)
M = point(3.5,5)
N = point(9,5)
P = point(6.2,6.1)
M L O equilateral
N P Q equilateral
N M R equilateral
S = intersection(line(N,O),line(L,R)) 
N S T equilateral
thickness(2)
draw(L,M,N)
thickness(0.5)
draw(N,P,Q)
draw(segment(M,R))
draw(segment(N,R))
draw(segment(O,L))
draw(segment(O,M))
draw(segment(R,Q))
draw(segment(P,L))
draw(segment(P,N))
draw(segment(M,P))
draw(segment(S,M))
draw($A$,L,0.2,20:)
draw($B$,M,0.2,180:)
draw($C$,N,0.2,0:)
draw($P$,P,0.2,180:)
draw($Q$,Q,0.1,180:)
draw($F$,O,0.2,180:)
draw($D$,R,0.2,0:)
draw($P_0$,S,0.1,45:)
draw($Q_0$,T,0.1,0:)
draw("\textbf{\Huge Case 2}",L,2.5,0:)
draw(segment(L,R))
draw(segment(N,O))
draw(segment(N,T))
mark(segment(M,P),simple,0.7)
mark(segment(Q,R),simple,0.7)
mark(segment(N,P),triple,0.7)
mark(segment(N,Q),triple,0.7)
mark(segment(P,Q),triple,0.7)
mark(segment(M,N),double,0.7)
mark(segment(M,R),double,0.7)
mark(segment(N,R),double,0.7)

Licencia

Yo, el titular de los derechos de autor de esta obra, lo libero al dominio público. Esto aplica en todo el mundo. En algunos países esto puede no ser legalmente factible; si ello ocurriese: Concedo a cualquier persona el derecho de usar este trabajo para cualquier propósito, sin ningún tipo de condición al menos que éstas sean requeridas por la ley.