Projection_Triangle

Triangle of projection-points

Let c be a circle concentric to the circumcircle of triangle ABC. Project orthogonally a point X of c onto the sides of ABC. The triangle A'B'C' of the projection-points has constant area, as X moves on the concentric circle. [Querret and Sturm, cited from Steiner, Werke, Bd 1, p. 15]
The particular case, where the concentric circle c coincides with the circumcircle, gives degenerate triangles (Simson lines).

1) switch to the pick-move tool (press CTRL+2)
2) pick-move point X
3) its shape changes but not his area (displayed in the number-object)
4) pick-move points A, B, C to change the shape of the basic triangle
5) the areas of ABC and A'B'C' change.
6) pick-move again X. The area of A'B'C' doesn't change
7) switch to selection-tool (press CTRL+1)
8) pick Y and drag to enlarge the outer circle. Repeat the experiments
9) give a proof of the proposition

For a generalization look at the file Projection_Polygon.html .

Triangle of projection-points

See Also