Nine-Point Circle

The nine-point circle of a triangle goes through the foot and midpoint on each side, as well as through each of the midpoints between the orthocenter and the vertices. Using CircleThrough, it is possible to compute the circles going through each triple of points and verify that they are all the same circle.

Define a triangle with arbitrary vertices.

Calculate coordinates for the three feet.

Calculate the circle going through those three feet.

Calculate coordinates for the three side midpoints.

Calculate coordinates for the orthocenter.

Calculate coordinates for the three midpoints between the orthocenter and triangle vertices.

Verify that the circles through the three sets of points are, in fact, all the same nine-point circle.

Draw the nine-point circle of a triangle.