feuerbach.md

Feuerbach's theorem states that the nine-point circle is tangent to the three excircles of the triangle as well as its incircle.

Start from Triangle

Put AB onto x-axis and C onto y-axis:

• The nine-point circle passes through three midpoints M_A, M_B, and M_C;
• The incircle passes through D, E and F, where AE = AF, BD = BF and CD = CE;
• One excircle passes through G, H and K, where AH = AK, BG = BK and CH = CG.

So we can get the equations of these three circles. Take the nine-point circle and the incircle as example, denote their equations as:

Eliminate y or x in either of the two equations by the radical line , then we get a quadratic equation about x or y.

Because we need to prove these two circles are tangent (i.e. they have only one common point), the remaining work is just checking if the discriminant of the quadratic equation is zero.

Here is the proof process.

Another approach is to calculate radii and distance:

• The incircle is inside and internally tangent to the nine-point circle if and only if ;
• One excircle is externally tangent to the nine-point circle if and only if .

Because all of |NI|, r_N, r_I and r_E contain square roots, which are too difficult to calculate, we need to rewrite them to simplify calculation:

Neither of the two approaches can tell us whether two circles are internally or externally tangent. However, the incircle is obviously inside and internally tangent to the nine-point circle, because the incircle is inside the triangle and the nine-point circle intersects all three edges; each excircle is abviously externally tangent to the nine-point circle, because each excircle is outside of the triangle.

Start from Incircle

If we use the coordinates mentioned here (proof of Theorem 1), it is possible to finish the proof by hand. Given an incircle:

and two vertices A(-a,0) and B(b,0) on x-axis, then C is . And the nine-point equation is:

The equation set of two circles (Eq. 1 and Eq. 2) has only one root (because of its zero discriminant), which proves the tangency of two circles.

An interesting thing is that Eq. 1 can also be an excircle of trangle ABC if we set . So the the tangency of nine-point circle and excircle is also proved.

The tangency can also be proved by calculating the nine-point circle's center:

and radius:

And the distance between the two circles' centers should be:

For the incircle case (), we have:

So we get , which means the incircle is inside and internally tangent to the nine-point circle.

For the excircle case (), we have:

So we get , which means the excircle is externally tangent to the nine-point circle.

Other proofs can be found here (Problem 7) and here.

Theorems related to the Feuerbach Point

Theorem 1 The circle through the feet of the internal bisectors I_AI_BI_C of a triangle ABC passes through the Feuerbach point.

Here is the computational proof starting from incircle.

Theorem 2 The Feuerbach point of a triangle ABC is the anti-Steiner point of the Euler line of the intouch triangle C_AC_BC_C with respect to the same intouch triangle C_AC_BC_C.

Here is the computational proof starting from incircle.

Other proofs of the two theorems can be found here (computational) and here (synthetic).

Theorem 3 Let F_A, F_B, F_C be the touch points of the nine-point circle with the A-, B-, C- excircles, respectively. The lines AF_A, BF_B, CF_C meet at X(12), the harmonic conjugate of the Feuerbach point F_I with respect to the incenter I and the nine-point center N. ¹

Here is the computational proof starting from incircle.

This theorem can also be proved by Monge's theorem (proposition 2).