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Fermat Point and Art
by Athanasia Soufari


On the catalogue of my favourite pages I found the web-page:
When I visited the site, I was caught by a painting published there. It impressed me with its geometric structure and its three color palette. But I was impressed even more by its title “Fermat Point”. I decided to construct the painting on the GeoGebra canvas using geometric construction tools, aiming to highlight its geometry and to describe its geometric properties.

Click on the picture below to open the interactive GeoGebra application.

GeoGebra: Fermat Point and Art

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To navigate from one slide to the other, just click on the links (highlighted words).


Fermat Point Definition

The Fermat-Torricelli Point is an inner point of a triangle ΑΒΓ with angles less than 120o with the property that the sum of its distances from the triangle vertices is minimal.

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Just like a painter’s canvas is the base of a painting, the “graphics window” on the GeoGebra software is the base of a geometric construction. In order to “play” with the colors, I created a quite big quadrilateral on the background, covering the graphics window.
There is the command Corner[ ]

  • 1st stage of the construction:

    I constructed the triangle ΑΒΓ, the equilateral triangles ΒΓΑ’, ΓΑΒ’, ΑΒΓ’, their circumscribed circles and the line segments ΑΑ’, ΒΒ’ and ΓΓ’.

    The intersection of the three circumscribed circles, which is identical to the intersection of the three line segments ΑΑ’, ΒΒ’ and ΓΓ define the Fermat Point F.

    Fermat by steps 1
  • 2nd stage of the construction:

    The painting is a synthesis of geometric shapes (polygons and circular segments). I tried to reproduce the painting with its palette (three colors) and the coloring rules set by the painter Suman Vaze.

    The circular segments where created by filling arcs (opacity 100%).

    In each circle we meet all three colors. When triangles form vertical angles they have the same color.

    The problem of the overlapping colors was solved by using layers. For instance, a circular segment that covers a part of a triangle should be placed on an upper layer, above of the triangle.

    Find attached a GeoGebra file (Fermat_by_steps.ggb) that explains the “coloring game”.

    Fermat by steps 2

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