Proof 5.31

Consider  where the vertices have the coordinates  and . The orthocenter, , of the triangle was found previously to be  and the center of the Nine Point Circle,, was found to be . We can determine the coordinates of the circumcenter through the following steps: Midpoints of the triangle sides found using the midpoint formula, . Midpoint of  Midpoint of  Midpoint of  Slope of the triangle sides are found using . Slope of  Slope of  Slope of  The slope of the line perpendicular to the triangle sides is found using . The lines of the perpendicular bisectors of the triangle sides can be found using the information from above and the point-slope line equation. We can find the circumcenter, , of  by finding the intersection point of two of the equations from above. We can substitute this x-value into one of the original equations to determine the y-coordinate of the intersection point. Based on the two calculations above, we know that the circumcenter of  is . Recall that the midpoint of a line segment can be found using the formula . Based on this, we can determine the midpoint of the line created by the circumcenter and the orthocenter. The midpoint is at .  If you recall, the center of the nine point circle is also located at  from the information given. Therefore, we can conclude that the nine point center is the midpoint of the segment from the orthocenter to the circumcenter. This means that the nine point center lies on the Euler Line. Q.E.D.