Medians Centroid Theorem (Proof without Words)
- Tim Brzezinski
Definition: A MEDIAN of a TRIANGLE is a segment that connects that triangle's VERTEX to the MIDPOINT of the SIDE OPPOSITE that vertex. Definition: 3 or more lines are said to be CONCURRENT LINES if and only if they intersect at exactly 1 point. (Their point of intersection is called the point of concurrency.) A triangle's 3 MEDIANS are ALWAYS concurrent. Their point of concurrency is called the CENTROID of the triangle. Did you know that the CENTROID of a triangle its center of gravity? It is. There is another interesting fact about a triangle's centroid you will soon discover after interacting with the applet below. The directions & investigation questions are displayed below the applet.
Questions. (BE SURE to drag/move vertices A, B, & C around during your pursuit of answers to these questions below!) 1) Is it ever possible for a triangle's CENTROID to lie OUTSIDE the triangle? If so, under what circumstance(s) will this occur? 2) Is it ever possible for a triangle's CENTROID to lie ON THE TRIANGLE ITSELF? If so, under what circumstance(s) will this occur? 3) If your answer for (2) was "YES", where on the triangle did point G lie? 4) Is it ever possible for a triangle's CENTROID to lie INSIDE the triangle? If so, under what circumstance(s) will this occur? 5) Click on the four checkboxes in the upper right hand corner (one after the other) to observe some pretty cool phenomena. 6) After clicking the "CHECK THIS OUT !!!" checkbox, be sure to answer the bold question that appears in red. Answer it in detail, trying to explain (as best you can) the phenomena you observe.