SimsonVariantLocus

Variations on Simson

The red line is an interesting, though somewhat complicated in its construction locus.
The idea is to rotate a fixed triangle t'=(A'B'C') in its circumcircle (center O) and rotate also another point (D) about O with a multiple (f=3) of the rotation velocity of t'. Then construct the projections (A'',B'',C'') of D on the sides of t' and get the circumcenter P of the triangle t''=(A''B''C''). The Geometric locus of P is the red line.

In the construction described below, the following elements are movable:
1) FREE movable: the triangle t= (ABC) and the points O and E.
2) ON CONTOUR movable: points A (on circle (OE) ) and D (on Line OD).
3) VARIABLE is also the number-object (factor f) which gives the angle-ratio <(EOD) over <(EOA').

To free move the points you switch to the selection-tool (Ctrl+1) and catch them.
To move-on-contour the points you switch to the selection-tool (Ctrl+2) and catch them.
To vary the factor f, click on it to select it and play with the up/down arrows on the keyboard.

Up to homothety, the shape of the locus depends
a) on the triangle-prototype t = (ABC),
b) on the factor f, and
c) on the distance |OD|.

The construction of a triangle, totating in its circumcircle is described in the file: RotatingTriangle.html .

For the construction of the multiple y=f*x of an angle, using a number-object (to represent f), look at the file: AngleMultiple.html .

Variations on Simson

See Also