Compass and Ruler C.a.R. Miquel's Pentagram - Dynamic Geometry

 

Take a pentagram ABCDE forming a convex pentagon FGHIJ and triangles AFJ, BGF, CHG, DIH, and EJI. Construct the circumcircles of triangles AFJ, BGF, CHG, DIH, and EJI. Then the five new points, K,L,M,N,P resulting from the intersection of two consecutive circumferences are concyclic (lie on the same circumference).

Instructions: Dynamic Geometry  (Requires Java 1.3 or higher and Java enable browser)

You can alter the figure above dynamically in order to test and prove (or disproved) conjectures and gain mathematical insight that is less readily available with static drawings by hand. To explore this theorem use the replay buttons above to move step by step (1-2-3-4-5) through the geometric construction:

  • To the Start (Step 1), Jump to next break (Step 2-3-4-5), To the end.

  • Manipulate the dynamic pentagram by dragging the points A,B,C,D,E at any step.

  • The cursor keys (left, right, up, down) move the picture. To give the keyboard focus to the applet, click into it.

  • The + and - key change the size magnification.

This page contains a C.a.R. interactive geometry applet by R. Grothmann. Please be patient while the applet loads on your computer. If you are using a dial-up connection, it may take a couple minutes. If you get a warning-security asking 'Do you want to trust the signed applet distributed by "Rene Grothmann"?'. Please click 'Always', and you will not be troubled again.
If you can't see the presentation above, check a more recent version of your browser. Alternately, it may be that your browser supports Java, but that it’s currently set to disable Java applets. Dynamic Geometry applet requires a Java Plug-in 1.3 or higher (More details at: http://www.java.com/en/download/help/index.jsp ).
 

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