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Worksheet 33.1 A Geometric Prelude

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This two-page worksheet was generously donated to the sample article by Dave Rosoff at a CuratedCourses workshop in August 2018. It has the default (skinny) margins.
It was known to Euclid, and probably earlier, that the midpoints of the sides of any quadrilateral all lie in the same plane (even if the vertices of the quadrilateral do not). In fact, these midpoints are the vertices of a parallelogram, as pictured in Figure 33.1.
Figure 33.1. The midpoints of the sides of a quadrilateral are the vertices of a parallelogram.
Figure 33.2. The sides of a triangle presented as vectors.
Figure 33.3. The medians of the triangle are \(\vec{M}_1\text{,}\) \(\vec{M}_2\text{,}\) and \(\vec{M}_3\text{.}\)
In this exercise, we’ll use vectors to show that the medians of any triangle (Figure 33.2) intersect at a point. Recall that medians are the lines connecting the vertices of the triangle to the midpoints of their opposite edges, as in the figure. We’ll do this in a few steps.


What is the value of \(\vec{A} + \vec{B} + \vec{C}\text{?}\)
Figure 33.3 from the previous page is reproduced for your convenience.
Figure 33.4. The medians of the triangle are \(\vec{M}_1\text{,}\) \(\vec{M}_2\text{,}\) and \(\vec{M}_3\text{.}\)


To show that the point \(P\) exists (as the common intersection of the \(\vec{M}_{i}\)), show that
\begin{equation*} \vec{A} + \frac{2}{3} \vec{M}_{3} = \frac{2}{3} \vec{M}_{2} = \fillinmath{\frac{2}{3} \vec{M}_{1}-\vec{C}}\text{.} \end{equation*}


If you have time, try to devise a vector proof of Euclid’s result presented at the beginning of the workshop. Recall that a parallelogram is a four-sided polygon whose opposite sides are parallel.


It’s possible to do interesting things with vector arithmetic in a coordinate-free way: we didn’t specify an origin, or any entries of any vectors in the examples.