This java applet draws the medians and
other area bisectors of a triangle. Click and drag within
the applet area to change the triangle. What
proportion of the triangle is the red deltoid? More information below. |

The blue lines above are the **medians** of the triangle.
Each one connects a vertex of the triangle to the mid-point of
the opposite edge. Each median divides the area of the triangle
in half. They intersect two-thirds of the way along their length
in the **centroid** of the triangle. The centroid is the
centre of mass (or center of gravity - depending on your spelling
and approach) of the solid triangle, so a solid triangle would
balance on any line through the centroid.

However, it is **not** true that all straight lines through
the centroid divide the area of the triangle in half. While the
medians do divide the triangle into two equal areas, other lines
through the centroid do not, and in the worst case (when the line
through the centroid is parallel to an edge) the tringle on one
side is ^{4}/_{9} of the area of the original
triangle, while the trapezium on the other side has an area ^{5}/_{9}
of the original triangle.

Some of the other lines which are **area bisectors** are
shown above in green. Together, their envelope is a deltoid,
shown in red, and any point inside the deltoid has three area
bisectors passing through it, while any point outside it has only
one. The curved edges of the detoid are segments of hyperbolae,
and its three vertices are the mid-points of the medians. Since
the proportions in the diagram are invariant under affine
transformations, the deltoid's area is a fixed proportion of the
area of the triangle. By taking a simple triangle, such as the
one with corners at (0,0), (0,1) and (1,0), it is not difficult
to find that the proportion is ^{3}/_{4}*log_{e}(2)-^{1}/_{2}
= 0.019860... If this seems small (less than a fiftieth the
original triangle), remember that the area of the triangle with
straight edges using the corners of the deltoid is only a
sixteenth of the area of the original triangle and the deltoid is
almost a third of this smaller triangle.

It is easy to construct a set of points met by all the medians
and other area bisectors. Indeed the intersection of the deltoid
with a median or another area biscetor is such a set. The
intersection of the deltoid and a median or other area bisector
is ^{1}/_{sqrt(2)} - ^{1}/_{2}
= 0.2071... of the length of that median or area bisector.

Some links:

See the Java code.

This page is based on discussions in the newsgroup geometry.puzzles in October and December 2001. Archives of these can be found here: 1 2 3 4 5.

Zak Seidov produced a couple of related pages on a 3,4, 5 triangle: Centroid and 3-points Deltoid Area

Rouben Rostamian produced a picture of the area of the deltoid

The following links to some of my pages are less directly relevant:

Triangular optical illusion

Area of a triangle (7 times)

Pythagoras's theorem in moving pictures

Circumnavigating Platonic polyhedra

Map projections of the world

Java and JavaScript examples

Henry Bottomley January 2002