Feuerbach's Theorem: Nine-Point Circle Tangent to Incircle

A visual demonstration of Feuerbach's Theorem, showing that the nine-point circle is tangent to the incircle at a single point called the Feuerbach point.

Keywords: Feuerbach's theorem, nine-point circle, incircle, tangent circles, Feuerbach point

Prerequisites: ninepointcenter-proof, incenter-proof · Difficulty: advanced

One of the most beautiful theorems in triangle geometry states that the nine-point circle is tangent to the incircle (and also to the three excircles). This remarkable result is known as Feuerbach's Theorem, named after Karl Wilhelm Feuerbach who proved it in 1822.

What Are These Circles?

Incircle: The largest circle that fits inside the triangle, touching all three sides. Its center is the incenter $I$ (where the angle bisectors meet), and its radius is $r$ (the inradius).

Nine-point circle: The circle passing through nine special points:

• The three midpoints of the sides
• The three feet of the altitudes
• The three midpoints of segments from vertices to the orthocenter

The nine-point circle has center $N$ (the midpoint of $OH$) and radius $R/2$ (half the circumradius).

Statement of the Theorem

Feuerbach's Theorem: The nine-point circle is tangent to the incircle. That is, these two circles touch at exactly one point.

The condition for two circles to be tangent is:

• External tangency: Distance between centers equals the sum of radii: $|NI| = r_1 + r_2$
• Internal tangency: Distance between centers equals the difference of radii: $|NI| = |r_1 - r_2|$

For most triangles, the incircle fits inside the nine-point circle, so they are internally tangent:

|NI| = \frac{R}{2} - r
    

The point of tangency is called the Feuerbach point.

Triangle and Incircle

We begin with triangle $ABC$. The incircle is the largest circle that fits inside the triangle, touching all three sides. Its center is the incenter $I$, where the angle bisectors meet.

The Nine-Point Circle

Next, let's construct the nine-point circle. The nine-point center $N$ is the midpoint of segment $OH$, where $O$ is the circumcenter and $H$ is the orthocenter. The nine-point circle passes through nine special points including the midpoints of the sides and the feet of the altitudes.

Two Circles

Now we have both circles: the incircle (center $I$) and the nine-point circle (center $N$). What is their relationship? They appear to touch at exactly one point!

The Tangency Condition

The distance between centers determines whether the circles intersect, are separate, or are tangent. Let's examine the radii.

• The inradius is $r$ (distance from $I$ to any side)
• The nine-point radius is $R/2$ (where $R$ is the circumradius)

For tangency, we need: distance between centers = difference of radii.

Feuerbach's Theorem

Feuerbach's Theorem (1822) states that the nine-point circle is tangent to the incircle. For most triangles, the incircle fits inside the nine-point circle.

The tangency condition is:

|NI| = \frac{R}{2} - r
    

This means the distance between centers equals the difference of radii.

The Feuerbach Point

The point where the two circles touch is called the Feuerbach point. It lies on the line $NI$, at distance $r$ from $I$ and distance $R/2$ from $N$. The Feuerbach point $F$ lies on both circles — it's the unique point where they touch.

A Beautiful Theorem

Feuerbach actually proved something even more remarkable: the nine-point circle is tangent not only to the incircle, but also to all three excircles! This gives four points of tangency total.

Feuerbach's Theorem connects circles defined by completely different constructions — angle bisectors (incircle) versus perpendicular bisectors and altitudes (nine-point circle). Yet they are always tangent! This is one of the most beautiful results in triangle geometry.

Conclusion

Feuerbach's Theorem shows a remarkable connection: the nine-point circle is tangent to the incircle, touching at a single point called the Feuerbach point. The relationship is:

|NI| = \frac{R}{2} - r
    

Extended Theorem

Feuerbach actually proved something even more remarkable: The nine-point circle is tangent not only to the incircle but also to all three excircles. This gives four points of tangency total.

Historical Note

Karl Wilhelm Feuerbach proved this theorem in 1822. It's considered one of the most beautiful results in Euclidean geometry, connecting circles defined by completely different constructions — angle bisectors (incircle) versus perpendicular bisectors and altitudes (nine-point circle). There's no a priori reason why they should be tangent, yet they always are!