CTK Insights

Tweet

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.

Problem

$\Delta ABC$ 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.