theorems

the following present the list of theorems and corresponding dynamic and step-by-step applets , that the author constructed to support a teacher’s training program, held in 2009, on dynamic geometry software. since the original applets (published here, in portuguese) no longer work, the contents in this page should be regarded as their reconstructions.

  • Napoleon’s theorem (3 applets, already finished)
  • Varignon’s theorem
  • Simson’s line
  • Euler’s line
  • Feuerbach’s theorem
  • Pappus’ hexagon

these applets are gathered in this this geogebra book.

each of the images below is a link to a step-by-step geogebra applet through which it is possible to move forwards and backwards in the construction, or watch how the construction is solved in a short animation. the “circular” points in the applets that bare movable are mentioned in the theorem’s description. The construction protocol buttons below some of the applets allows one to see how the constructions were accomplished.


Napoleon’s theorem

the discovery of this theorem is attributed to Napoleon Bonaparte (1769-1821), who was very much interested in mathematics, particularly, geometry, but the fact that Napoleon was actually capable of formulating this theorem is debatable, as Coxeter and Greitzer (2008: 63) denote.

according to this theorem, the centres of the equilateral triangles drawn outwards to the edge of any triangle outline, themselves, another equilateral triangle, which has been identified as the outer Napoleon triangle.

in the applet below, the vertices of the triangle [ABC], in black, are movable. The external triangles were drawn in blue dashed lines and their centres outline the blue equilateral triangle [O1O2O3].

analogously, it has been concluded that the centres of the equilateral triangles drawn inwards to the edge of any triangle outline the Napoleon’s innner triangle, which is also equilateral. in the construction, this second triangle [M1M2M3] has been drawn in red.

the perpendicular bisectors of the edges of both Napoleon triangles intersect in the same point, their common circumcentre, X.

lines AO2, BO3 and CO1 (in green) intersect in the same point, M.

D, E and F are the outmost vertices of the outer Napoleon triangles. lines DO1, EO2 and FO3 (in blue) intersect in the same point, R, which is the circuncentre of [ABC].

the circumcircles of the external equilateral triangles intersect in the first Fermat point. Segments [AE], [BF] and [CD] have equal length and intersect in the same point. This point, also known as Fermat’s point, Torricelli’s point or first isogonic point, was discovered by Torricelli (1608-1647), after a challenge proposed by Fermat (1601-1665), and is defined as the point in the plane whose sum of the distance to three other points A, B and C is the least possible.

If this point is inside triangle [ABC], the angles between the segments connecting F to a, b and C are equal to 120º. this property applies only to triangles in which none of the internal angles exceeds 120º.

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