The Nine-Points Centre
The mid-points of the sides, L, M, N, and the feet of the perpendiculars, D, E, F, are concyclic. The circle through them is called the nine-points circle, since it also passes through the midpoints of the upper parts of the altitudes AH, BH, CH. The name is convenient but a misnomer since in an isosceles triangle the 9 points reduce to 8, in a regular triangle to 6, and in a right-angled triangle to only 5. The centre of this circle is the nine-points centre.
The reflection of a line through a vertex in the bisector is called its isogonal line. The line joining the points where two isogonal lines meet the circumcircle is parallel to the side opposite the vertex concerned. The isogonals of three concurrent lines are themselves concurrent. These pairs of points are called isogonal conjugates of each other. The isogonal conjugate of O is the orthocentre H, since if the diameters AO, BO, CO meet the circumcircle at A'B'C' and the isogonals meet the circumcircle at say A''B''C'' then AA''A' is a right angle (angle in semicircle) and since A'A'' is parallel to BC then the isogonal AA'' is perpendicular to BC, i.e.it is an altitude.
The circumcentre and orthocentre O and H are isogonal conjugates, the altitudes and circumdiameter issued from the same vertex are isogonal lines, i.e. angle CAO = angle BAH etc, or angle OAB = angle HAC etc. For any such pair of isogonal lines AU, AV if UU', VV' are the perpendiculars from U and V to AB and UU'', VV'' are the perpendiculars to BC then the four points U', U'', V', V'' are concyclic, U'U'' is perpendicular to AV, and V'V'' is perpendicular to AU, and the circle is centred at the midpoint of UV. Thus the nine-points centre is the midpoint of the line OH.
The median centre G also lies on the line OH, which is known as the Euler line of the triangle. In a right angled triangle it is the median from the right angle. In an isosceles triangle it coincides with the axis of symmetry. OH² = R².(1 − 8.cosa.cosb.cosg).
H is the centre of similitude of the circumcircle and the nine-points circle. Thus the nine-points circle bisects every line from H to the circumcircle. AH² = 4.R² − a², etc.
If XY is any given line and A', B', C' are the feet of the perpendiculars from A, B, C to the line XY then the perpendiculars A'A'', B'B'', C'C'' from A' to BC, from B' to CA and from C' to AB are concurrent. This meeting point is called the orthopole of the line XY with respect to the triangle ABC. The orthopole of a circumdiameter lies on the nine-points circle.
K.W.Feuerbach, Erlangen 1822, proved that the nine-point circle touches the incircle. The point of contact is called the Feuerbach point of the triangle. In the case of a regular triangle, exceptionally, the two circles coincide, so all the points on the circle are 'Feuerbach points' in this case. The nine-points circle also touches all three excircles.