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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