Proving the construction of Fermat Point is valid: We have to prove the three lines constructed are concurrent. The red and blue triangle are congruent by S.A.S. , and hence the angles in the same segment are the same, which shows that there are two cyclic quadrilaterals. Thus the last four points are also concyclic, and by angle in the same segment the last line is a straight line.
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Date
11 March 2006 (original upload date)
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fermat_point_proof.euk
frame(-2.8, -6, 8.2, 7.5) B C A triangle(6, 75:, 35:) C B P equilateral A C Q equilateral B A R equilateral s = segment(A, P) t = segment(B, Q) u = segment(C, R) c = circle(B, A, R) d = circle(A, C, Q) e = circle(C, B, P) F = intersection(line(A, P), line(B, Q)) color(green) draw(c) color(cyan) draw(d) color(blue) draw(t) draw(segment(A, Q)) draw(segment(A, B)) mark(segment(A, B), simple) mark(segment(A, Q), double) mark(B, A, Q, simple, 1.2) color(red) draw(u) draw(segment(A, R)) draw(segment(A, C)) mark(segment(A, R), simple) mark(segment(A, C), double) mark(R, A, C, simple, 0.8) color(black) draw(s) draw(segment(B, R)) draw(segment(C, Q)) draw(C, B, P) draw(e, dashed) draw(F, dot, 1.3) mark(segment(B, R), simple) mark(segment(C, Q), double) mark(segment(B, C), triple) mark(segment(B, P), triple) mark(segment(C, P), triple) mark(C, R, A, double, 1.5) mark(Q, B, A, double, 1.5) mark(A, Q, B, triple, 1.5) mark(A, C, R, triple, 1.5) mark(B, F, P, dot, 0.8) mark(B, C, P, dot, 0.8) label(F, 0.5, 0:) label(A, 0.5, 90:) label(B, 0.5, 225:) label(C, 0.5, 330:) label(P, 0.3, 270:) label(Q, 0.3, 0:) label(R, 0.3, 90:)
Paste the resulting code in the following en:TeX file and compile it into eps.
\documentclass{article} \usepackage{pstricks} \usepackage{color} \begin{document} \pagestyle{empty} \colorbox{white}{ %Paste the code here } \end{document}
Import the eps file using en:Scribus. (Remember to install en:ghostscript also and configure the path to ghostscript correctly in Scribus's Preferences)
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Proving the construction of Fermat Point is valid: We have to prove the three lines constructed are concurrent. The red and blue triangle are congruent by S.A.S. , and hence the angles in the same segment are the same, which shows that there are two cycl