CTK Insights

07 Jun

Dynamic Software as Serendipity Enhancement

Several years ago I wrote a page with a Java illustration to a solution of problem 4 from the 1995 British Mathematical Olympiad. Earlier today I happened on that page and noticed that the solution refers to point L that is not marked in the applet. Since the time I wrote that page, I moved to another computer and to using GeoGebra instead of writing Java applets, so much so that I do not even have my Java development environment on my present computer. To resolve the issue I decided to put together a GeoGebra applet with L properly marked and replace the old Java applet. This took probably a couple of minutes. However, this is not what made my day.

At the end of the old page I observed that the configuration in the problem had additional features and pointed to two of them. Now, GeoGebra makes it easy to experiment - form and verify hypothesis, or just plain count on serendipitously stumbling on a dormant feature. It was a sin not to try. The configuration in the problem definitely had many more features waiting to be discovered. Simple though all it was, I am satisfied to have found some.


\Delta ABC has right angle at C. The internal bisectors of angles BAC and ABC meet BC and AC at P and Q respectively. The points M and N are the feet of the perpendiculars from P and Q to AB. Find angle MCN.

The angle was found to be 45^{\circ}. Checking the "Hint" box in the GeoGebra applet below will show the essential steps of the proof.

This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com

Checking the "Extra" box will suggested a few more properties: \angle MCN is not the only angle in the diagram that equals 45^{\circ} (e.g., \angle ADN and \angle CMD); \angle NLM=90^{\circ}; some intersections (N,D,L,E,M) are concyclic; there are several similar triangles (e.g., \Delta ALN and \Delta NLD).

There are probably other properties. Should you find any, please let me know.

4 Responses to “Dynamic Software as Serendipity Enhancement”

  1. 1
    Joshua Zucker Says:

    I get an error "ClassNotFoundException geogebra.GeoGebraApplet" on this page.

  2. 2
    Joshua Zucker Says:

    Never mind, I figured out how to cure it by changing some Java permissions in my browser.

  3. 3
    Joshua Zucker Says:

    You have a typo in the text at the bottom, <MAN when you mean <MCN.

  4. 4
    admin Says:

    Joshua, thank you. I just discovered a pluggin for WordPress that permits to embed Java, GeoGebra - anything at all - in a blog. Couldn't help it but try.

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