The Nine-Point Circle

It was Karl Wilhelm Feuerbach, a German geometer who recognized 6 specific points in a context of a given triangle that are all concyclic i.e. there esixts a unique circle passing through these 6 points and hence called a Six-Point Circle. The points through which this six-point circle passes are;

three midpoints of the sides of the triangle and other three points are the feet of the altitudes of the triangle. But before Feuerbach, there was a mathematician Olry Terquem who proved the existence of such circle with three extra significant points called the Euler's Points which are actually midpoints of the line segments from each vertex of the triangle to orthocentre of the triangle and that's why Terquem was the first to call it a Nine-Point Circle. This nine-point circle is also known as Euler's Circle or Terquem's Circle or Feuerbach's circle. So we can say that corresponding to each given triangle, there exist nine significant points, three of them are the midpoints of the sides of the given triangle, three the feet of altitudes of the triangle and rest three the midpoints of the line segments joining the vertices to the orthocentre of the triangle, (Euler's Points) all being concyclic and the circle passing through these nine points called a nine-point circle.

■ Spectaculars (Features/Properties) of Nine-Point Circle;

● For each triangle, there exists a unique nine-point circle called Nine-Point Circle Theorem.

● A line passing through all important central points viz. Orthocenter (H), Centroid (G), Circumcenter (O) etc. in a given triangle which is not equilateral is called Euler Line.

● The center of this circle is a point lying on Euler line at the exact midpoint between the triangle's orthocenter and circumcenter.

● The radius of this circle is exactly half the radius of the circumcircle of the given triangle.

● The nine-point circle is surprisingly a tangent internally to the triangle's incircle and tangent externally to its three excircles and this statement is called Feuerbach's Theorem.

■ Note : Well although this name is nine-point circle but when we deal with equilateral triangles there are remaining fewer such points.

👉 For equilateral triangles, due to the symmetry, the orthocenter, circumcenter, centroid, and nine-point center all merge into a single central point giving rise to only 6 such points.

👉 On the other hand for isosceles triangles, the altitude, median, and perpendicular bisector drawn from the vertex angle to the base are all the exact same line due to the symmetry and this results, foot of the altitude on the base and the midpoint of the base becoming the same point, so our nine-point circle passes through only 7 distinct points instead of 9.