## Concurrent Medians Construction

An applet to show that the three medians of a triangle all meet in a single point.  The median of a triangle is a line  from one vertex to the midpoint of the opposite side.

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The three vertices of the triangle above can each be dragged to show that the three medians still meet in a single point when we look at other triangles

When a mathematician sees a behavior like this that works with all triangles (or at least with a bunch of examples that we have looked at) the suspicion is that there must be a structure that helps us prove that it must always happen.

For this construction we first want to establish a lemma, that if we take three copies of a triangle, two in a row and the third flipped to fill in, that the base line forms a straight line.  This uses the fact that the sum of the angles of a triangle adds up to 180 degrees, which is the angle of a straight line.  The lemma check box gives us this construction, with the point n draggable.  In particular, note that the point Q stays on the median for all locations of N.

From this we build to a final construction, putting 9 small triangles together into a single large triangle with each side three times as long.  consider the center point of any line parallel to an edge.  That point must lie on the corresponding median.  Thus the central point of the construction must lie on all three medians.

Created with GeoGebra
GeoGebra is a GNUed software package for mathematics visualization.  The home for the applications is  http://www.geogebra.org.