## Generalizations of a theorem of Sylvester

**NOTE**: Please WAIT while the applets below load. If a security pop-up menu appears in your browser, please choose **RUN/ALLOW** to let the applets run properly. *It is completely safe & can be trusted*. If you have the very latest Java on your PC or Apple Mac, and experience problems with the applets loading, please go *here* for additional information on Java settings that should resolve the issue.

The Euler line of a triangle is mostly valued, not for any practical application, but purely as a beautiful, esoteric example of post-Greek geometry. However, the following theorem by the British mathematician James Joseph Sylvester (1814-1897) involves an interesting application of forces that relate to the Euler line (segment).

**Theorem of Sylvester**: The resultant of three equal forces *OA*, *OB* and *OC* acting on the circumcentre *O* of a triangle *ABC*, is the force represented by *OH*, where *H* is the orthocentre of the triangle.

The result generalizes to forces acting similarly on *any point* as follows and as shown in the interactive sketch below:

**Generalization**: The resultant of three forces *PA*, *PB* and *PC* acting on any point *P* of a triangle *ABC*, is the force represented by 3*PG*, where *G* is the centroid of the triangle.

Drag any of the red points *A*, *B*, *C* or *P* in the sketch below.

####

A generalization of Sylvester's theorem to any point in a triangle

Can you explain why the result is true? (I.e. prove it?) First try yourself, but if stuck, click on the button in the sketch to show some constructions that might assist you in using the parallelogram law of forces to prove the result.

The result generalizes further to any point in a quadrilateral as follows and as shown in the interactive sketch below:

**Quadrilateral generalization**: The resultant of four forces *PA*, *PB*, *PC* and *PD* acting on any point *P* of a quadrilateral *ABCD*, is the force represented by 4*PG*, where *G* is the centroid of the quadrilateral.

Drag any of the red points *A*, *B*, *C*, *D* or *P* in the sketch below.

####

A generalization of Sylvester's theorem to any point in a quadrilateral

Can you explain why this result is true? (I.e. prove it?) Can you generalize further to other polygons or to 3D? See *Sylvester's theorem for a tetrahedron*.

See my paper *Generalizing a problem of Sylvester* for more details.

This page uses **JavaSketchpad**,
a World-Wide-Web component of *The Geometer's Sketchpad.*
Copyright © 1990-2011 by KCP Technologies, Inc. Licensed only for
non-commercial use.

Updated by Michael de Villiers, 15 September 2012.