Hexagon

This project is an example of Pascal's Mystic Hexagon problem.

Graph the parabola y = x² on a sheet of graph paper. We are going to make a hexagon (a six-sided figure) by picking 6 points on the parabola and connecting the points with line segments. For the six points, use the following x - coordinates. For each of the points you need to find the corresponding y - coordinates and label these points on your graph. For point A, let x = -3; for point B, let x = -1; for point C, let x = 0; for point D, let x = 2; for point E, let x = 3; and for point F, let x = 4. Once you have labeled the points A-F on your graph, connect point A and point B with a line segment, connect point B and point C with a line segment, connect point C and point D with a line segment, connect point D and point E with a line segment, connect point E and point F with a line segment, and connect point F and point A with a line segment. You should have a 6 sided figure called a hexagon.

We now are going to look at the intersection of "opposite'' lines. For the first case, you need to find the intersection of the line through point A and point B with the line through point D and point E. To do this, you need to find the equations of these lines. Once you have found the equations for these two lines, find their point of intersection (or in other words, solve this system of equations). On your graph extend these two line segments and mark the point of intersection on your graph. Label this point of intersection i on your graph.

For the next case, we want to find the intersection of the line through point B and point C with the line through point E and point F. To do this, you need to find the equation of the line through point B and point C and the equation through point E and F. Once you have found these equations for these two lines, find their point of intersection. On your graph extend these two line segments and mark the point of intersection on your graph. Label this point of intersection ii on your graph.

For the last case, we want to find the intersection of the line through point C and point D with the line through point F and point A. To do this, you need to find the equations of these two lines. Once you have found these equations for these two lines, find their point of intersection. On your graph extend these two line segments and mark the point of intersection on your graph. Label this point of intersection iii on your graph.

You should now have on your graph the three points of intersections marked that you found from solving the system of equations. If you have graphed your lines and parabola neatly enough, these three points should look like that they are all on the same line. To verify that they are all on the same line, find the slope between the first (i) and second (ii) point, the second (ii) and third point (iii) and the first (i) and third (iii) point. What should be true about the slopes if these points are all on the same line?