the list below presents the theorems and corresponding dynamic, step‑by‑step GeoGebra applets that I originally created in 2009 to support a teacher‑training programme on dynamic geometry software. The first versions of these applets were published online in Portuguese and, although that site is no longer available, the contents here should be regarded as their reconstructions.
these applets are also gathered in this this GeoGebra book. the images in the links below link to a step‑by‑step construction in which you can move forwards and backwards, or watch a short animation of the solution; movable “circular” points are mentioned in each theorem’s description, and the construction protocol buttons (where present) make it possible to inspect the construction process in detail:
- Napoleon’s theorem
- Varignon’s theorem
- Simson’s line
- Euler’s line
- Feuerbach’s theorem
- Pappus’ hexagon
Napoleon’s theorem
the discovery of this theorem is traditionally attributed to Napoleon Bonaparte (1769–1821), who is said to have had a strong interest in mathematics, particularly in geometry. However, whether Napoleon actually formulated the theorem himself is debatable, as Coxeter and Greitzer point out. [1, p. 63]
according to the theorem, if equilateral triangles are constructed externally on the sides of any triangle, then the centres of these equilateral triangles form another equilateral triangle, usually called the outer Napoleon triangle. In the applet below, the vertices of the black triangle [ABC] are movable; the external equilateral triangles are drawn with blue dashed lines, and their centres form the blue equilateral triangle [O1O2O3].
analogously, if equilateral triangles are constructed internally on the sides of the same triangle, the centres of these triangles form a second equilateral triangle, known as the inner Napoleon triangle. In the construction, this second triangle [M1M2M3] is drawn in red. The perpendicular bisectors of the sides of both Napoleon triangles intersect at a common point X, which is the shared circumcentre of the two triangles.
the lines AO₂, BO₃ and CO₁ (in green) are concurrent and meet at the point M. Points D, E and F are the outer vertices of the external equilateral triangles, and the lines DO₁, EO₂ and FO₃ (in blue) intersect at the point R, which is the circumcentre of triangle [ABC].
the circumcircles of the external equilateral triangles intersect at the first Fermat point of triangle [ABC]. The segments [AE], [BF] and [CD] all have the same length and are concurrent; they also meet at this same point, the Fermat point [2], Torricelli’s point, or first isogonic point, originally studied by Torricelli (1608–1647) after a challenge proposed by Fermat (1601–1665). it is defined as the point in the plane where the sum of the distances to the three vertices A, B and C is minimal.
when this point lies inside triangle [ABC], the angles between the three segments joining it to A, B and C are all equal to 120º. This configuration occurs precisely when none of the interior angles of triangle [ABC] is greater than or equal to 120º.
references:
- Coxeter, H. S. M., and S. L. Greitzer. Geometry Revisited. Washington, DC: Mathematical Association of America, 1967.
- Weisstein, Eric W. “Napoleon’s Theorem.” MathWorld – A Wolfram Web Resource. Accessed March 28, 2026. https://mathworld.wolfram.com/NapoleonsTheorem.html.
- Weisstein, Eric W. “Fermat Points.” MathWorld – A Wolfram Web Resource. Accessed March 28, 2026. https://mathworld.wolfram.com/FermatPoints.html.
moving geometry 