The Brocard Disc

Please enable Java for an interactive construction.

The above applet calculates the positions of various centres of a reference triangle ABC, and additionally plots these relative to the disc of diameter OSym (the disc enclosed by the Brocard circle). It is offered to lend weight to the hypotheses advanced (generally without mathematical proof) in section 4 of my report "Applications of the GPAT to triangle geometry", which you can find more about here.

The points are as follows:

Icon Object Label

Black point Circumcentre O

Dark red point Symmedian point Sym

Red point Centroid G

Blue point Gergonne point Ge

Green point Incenter I

Orange point Mittenpunkt M

Grey points Isodynamic points S and S'

Grey Circle Circle of diameter OSym (Brocard circle) D_OSym

Red Circle Circle of diameter OG D_OG

Purple Circle Parry circle P

The triangle vertices, points A,B and C, can be moved and the other points will be re-calculated as a result; the disc can be repositioned for convenience- the circumcentre can be moved freely whilst the Symmedian point is naturally constrained to the circle boundary.

Applet development in Cinderella

Icon	Object	Label

Black point	Circumcentre	O
Dark red point	Symmedian point	Sym
Red point	Centroid	G
Blue point	Gergonne point	Ge
Green point	Incenter	I
Orange point	Mittenpunkt	M
Grey points	Isodynamic points	S and S'

Grey Circle	Circle of diameter OSym (Brocard circle)	D_OSym
Red Circle	Circle of diameter OG	D_OG
Purple Circle	Parry circle	P