HISTORY

The credit of discovering nine point circle can’t be given to any one mathematician, there were several independent studies done on nine point circle starting from the beginning of nineteenth century. Karl Wilhelm Feuerbach was one of the German mathematicians who discovered six point circle (considering mid-points of sides and feet of altitudes).

Olry Terquem, a French mathematician was the first to prove that the six point circle also contains the midpoints of the line segments connecting each vertex to the orthocenter of the triangle and hence is a nine point circle. He also gave a new proof of Feuerbach’s theorem that the nine-point circle is tangent to the incircle and excircles of a triangle.

Other names of nine-point circle are:

1. Feuerbach’s circle,
2. Terquem’s circle,
3. thesix-points circle,
4. thetwelve-points circle,
5. Euler’s circle,

EULER’S LINE

A line passing through Centroid(G), Orthocentre(H) and Circumcentre(O) is known as Euler line.

 

 

DEFINITION:

• Nine Point Circle: A triangle which passes through mid points of sides of triangle, feet of altitudes and mid point of vertices and orthocentre is known as Nine-Point Circle.

 

 

 

 

PROPERTIES

• Nine point centre lies on Euler’s line.

• The center N of the nine-point circle bisects a
  segment from the orthocenter H to the circumcenter

 

• The nine-point center N is one-fourth of the way along the Euler linefrom the centroid G to the orthocenter H

 

 

 

 

• Diameter of nine point circle = FR = PD = EQ

 

 

 

 

 

• The radius of a triangle’s circumcircle is twice the radius of that triangle’s nine-point circle.
  

 

 

 

 

 

• Nine point circle is tangent to incircle and excircles of a triangle.

 

 

 

 

 

• A nine-point circle bisects a line segment going from the corresponding triangle’s orthocenter to any point on its circumcircle.

 

 

 

 

• The nine-point circle of a reference triangle is the circumcircle of both the reference triangle’s medial triangle( triangle with vertices at the midpoints of the sides of the reference triangle) and its orthic triangle ( triangle with vertices at the feet of the reference triangle’s altitudes)