NINE POINT CIRCLE

Definition Of Nine Point Circle

In an arbitrary triangle, the circle on which lie the 3 midpoints of the sides, the 3 feet of the altitudes and the 3 points which are the midpoints of the segments joining the orthocenter to the vertices of the triangle, is known as the nine-point circle.

More About Nine Point Circle

The nine-point circle is also known as Feuerbach's circle, Euler's circle, or Terquem's circle.
The radius of any nine-point circle is half the length of the radius of the circumcircle of the corresponding triangle.

Video Examples: Nine point circle

Example of Nine-point Circle

The above diagram shows the nine-point circle. In triangle ABC, the midpoints of the three sides M₁, M₂, M₃, the feet of the altitudes A₁, A₂, A₃, and the midpoints of the segments between the orthocenter and each of the three vertices, E₁, E₂, E₃, lie on a circle.

Solved Example on Nine Point Circle

Ques: If the radius(r₁) of the circumcircle of the triangle ABC is 8 cm, then find the radius(r₀) of the given nine-point circle.

Choices:

A. 5 cm
B. 3 cm
C. 4 cm
D. 6 cm
Correct Answer: C

Solution:

Step 1: The radius of any nine-point circle is half the length of the radius of the circumcircle of the corresponding triangle.
Step 2: r₀ =1/2 r₁ [From definition.]
Step 3: r₀= 1/2 × 8 [Substitute r₁ = 8.]
Step 4: = 4 cm [Simplify.]