Tiffany N. KeysŐ Essay 3:
Trisection
of the Area of a Triangle
Given
a triangle ABC, find a point D such that line segments AD, BD, and CD trisect
the area of the triangle into three regions with equal areas. Define D and
prove that the triangle is divided into three regions of equal area. Show a
construction for finding D
Triangle
BCD and the segment CE which is a median of triangle ABC were constructed.
The
area of triangle BCD = 1/2bh.
Therefore the area of triangles BCA, CAD, and BAD will be 1/3 the area
of triangle BCD, or 1/3 × 1/2bh = 1/6bh.
In
order to trisect the area of triangle BCD, I constructed a point A such that
segment AE that was 1/3 the length of segment CE.
The
point A that divides the median of a triangle in the ratio of 2 : 1 is called
the centroid of a triangle.