Fagnano Gergonne Locus Morley Rotation Squares Home Web pages Pictures Geometrikon

The following figures show some drawings created by EucliDraw. All of them were build in less than one minute after, of course, having spend some time to learn the use of tools. After the pictures follow some remarks about their relation with web pages, and how EucliDraw can help to build up a site with geometric content.
The Geometrikon is a collection of various documents, created in lectures, exercises, experiments and seminars on the use of the application.

## Fagnano

Here is a view of the figure used in a solution of the Fagnano Problem: Given the accute-angled triangle ABC, find the position of the inscribed triangle DEF, that has the least possible perimeter.

## Gergonne

Here is a view of the figure illustrating the Universal Property of the Gergonne Point of a triangle: Every conic inscribed in a triangle ABC is the image of the incircle under an Homography that fixes the vertices of the triangle and maps the Gergonne point G to an arbitrary point P. The role of P is indicated by the figure.

## Locus

Here is a view of a simple dynamic Geometric-Locus-Curve: Point A moves on a fixed circle. C is its projection on a fixed line. B is the projection of C on the diameter OA. As A moves on the circle, B describes the nice curve shown. Such geometric loci are instantly produced by clicking (with the appropriate tool) on A and B. The locus is dynamic. Its shape changes if the circle or the line are modified.

## Morley

Here is a view of Morley's Theorem: Construct the trisectors of the angles of a triangle ABC and consider the three intersection points D, E, F of these trisectors, as shown. The triangle DEF, having its vertices at these points, is equilateral. This is a nice case of how the exact dynamic drawing can lead to suggestions of new theorems.

## Rotation

Here is an interpretation of a rotation as product of two reflexions. The rotation is about an angle that is double the angle of the "mirors". The rotation's center is the intersection point of the two mirors.

## Squares

Here is a view of two (out of six) squares that can be inscribed in a generic quadrangle. The two squares have their opposite vertices on opposite sides (or their extensions) of the generic quadrangle. The other four, not shown, have their opposite vertices on adjacent sides of the quadrangle. The figure is constructed using a "User-Tool" i.e. a tool programmed (or scripted) by the user and compiled by the compiler included in the programming environment of EucliDraw. After having the tool, the construction of the two squares is a matter of clicking with the mouse on the given quadrangle. But to construct the tool ... is another nice story, involving knowledge of geometry and programming.

Here is also a picture, produced with the standard tools of EucliDraw, showing all six inscribed squares in a generic quadrangle ABCD.

## Web-Pages produced from EucliDraw-documents

Euclidraw can translate his documents to web-pages, automatically, by pressing a button. Of course the web-pages are static and deprived of the dynamic properties of the original EucliDraw documents. But this feature can be useful for presentations and introductions to some subjects. Then a more detailed discussion can use the original documents. In this spirit was produced the collection of geometric pages Geometrikon.

## Pages split into pieces

Here is another possibility to produce pages for your web site. The application can produce html-pages that are large images, split into pieces, so that they load fast when someone visits your site. The example here is a page containing text and the image of Pappus theorem, generalizing the well known theorem of Pythagoras. It is the same figure used in our logo. Click on the previous link to go to that page. You don't notice it when you see it, but the page (text + image) is split in 4 x 4 = 16 small (png) pictures. The browser starts working immediately (so you get a feedback), and rebuilds the page by placing the pieces in their right places. The whole system is produced by EucliDraw with a button-click. The same system is used for the pictures, contained in the pages of the Gallery of figures Geometrikon.

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