Eight Point Circle Theorem: Quadrilateral. Interactive Geometry Software.

Hint to
interact with the figure below: Click the
red button ()
on the figure to start the animation.
Move the points A, C, and D to explore
the figure. Press P and click the left mouse button to start the step by step
construction, help.

Proposition
Given a quadrilateral ABCD with perpendicular diagonals AC and BD. The four midpoints E,F,G, H
of each sides, and the four feet E',F',G',H' of perpendiculars from the
midpoints to the opposite sides all lie on a circle called the eight-point circle.

The center O of the eight-point circle is the centroid or
gravity center of quadrilateral ABCD.

The eight point circle was first discovered by Louis Brand of Cincinnati in 1944. This theorem can be useful in proving the far more famous
nine point circle theorem.
Reference: The American Mathematical Monthly, Vol. 51, No. 2 (Feb., 1944), pp. 84-85

Dynamic Geometry: You can alter the figure
dynamically in order to test and prove (or disproved)
conjectures and gain mathematical insight that is less
readily available with static drawings by hand.

This page uses the
TracenPoche
dynamic geometry software and requires
Adobe Flash player 7 or higher.
TracenPoche is a project of Sesamath, an association of French
teachers of mathematics.

Instruction to explore the
dynamic figure:

Animation. Click the red
button
to start/stop animation

Manipulate. Drag points A
, C, and D to change the figure.

Step-by-Step construction.
Press P and click the left mouse
button
on any free area of the figure
above to show the
step-by-step bar and click 'Next
Step' button ()
to start the construction step-by-step:

Hide the step-by-step bar by
using again the combination P +
click left mouse.