NINE POINT CIRCLE

In mathematics, the nine-point circle is a circle which can be constructed for any given triangle. It is named nine – point circle because it passes through nine significant points defined from a triangle.

The Nine point circle was discovered by Olry Terquem. He was a French mathematician known for his excellent work in geometry. He was the founder of 2 Scientific journals about geometry and the history of mathematics.

It is also known as Euler’s circle, Terquem’s circle, Feuerbach’s circle, and Mid circle.

In simple words, it is a circle that can be constructed for any given triangle and it passes through nine significant points.

The center of the constructed circle lies at the nine-point center of the triangle, and the radius of the nine-point circle is exactly half of the circumradius of the triangle that is: –
N = 1/2R
The center of the nine-point circle is known as the nine-point center and is usually denoted by N.

The nine points of the circle are:

• Midpoint of each side of the triangle ABC
• Foot of each altitude
• Midpoint of the line segment from each vertex of the triangle to the orthocenter (orthocenter is the point where three altitudes meet and these line segments lie on their respective altitudes).

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Properties of Nine Point Circle

1. The Nine-Point Center N lies on the Euler Line of the Triangle. The Euler line is the line passing through the orthocenter, the circumcenter, and the centroid of a triangle.
2. The tangents to the Nine-Point Circle at the midpoints of the sides of the triangle form a triangle that is similar to the orthic triangle.
3. The sides of this triangle are parallel to the triangle formed by the midpoints of the triangle.
4. The Nine-Point Circle of the triangle “touches” (touches here means “tangent”) the incircle(A circle inside a triangle that is tangent to each side) and the three excircles(A circle outside a triangle that is tangent to one of its sides and tangent to the extensions of the other two).

Note: – Every triangle has three definite excircles, each tangent to one of the triangle’s sides.

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How can we Construct  a Nine Point Circle ?

1. Firstly, draw a triangle on the plane surface.
2. Now, mark the midpoints of the sides of the drawn triangle.
3. Draw an altitude to the opposite side from each of the vertices and mark the points.
4. At last, mark the common point where the altitudes intersect.

Our nine point circle will look like this after following the given steps: –

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Proof of Existence of Nine Point Circle

As we all know, the reflection of the orthocenter (orthocenter is the point where three altitudes meet and these line segments lie on their respective altitudes) to the sides and to the midpoints of the triangle’s sides lie on our circumcircle.

Thus, by considering the homothety( It means we can use properties of similarity) centered at nine point center N with ratio.

It states that the circumcircle of the triangle ABC to the nine-point circle, and the vertices of the triangle to its Euler points(In a triangle, the Euler points are the midpoints of the segments joining the orthocenter to the vertices, Euler points lies on a nine-point circle).

Hence, the existence of the 9-point circle is proved.