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!

% 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)

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