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 .

Projection_Polygon.html

Simson.html

SimsonDiametral.html

SimsonGeneral.html

SimsonGeneral2.html

SimsonProperty.html

SimsonProperty2.html

WallaceSimson.html

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