In thegeometry oftriangles, theincircle andnine-point circle of a non-equilateral triangle are internallytangent to each other at theFeuerbach point of the triangle. The Feuerbach point is atriangle center, meaning that its definition does not depend on the placement and scale of the triangle. It is listed as X(11) inClark Kimberling'sEncyclopedia of Triangle Centers, and is named afterKarl Wilhelm Feuerbach.^[1][2]

Feuerbach's theorem, published by Feuerbach in 1822,^[3] states more generally that the nine-point circle is tangent to the threeexcircles of the triangle as well as its incircle.^[4] A very short proof of this theorem based onCasey's theorem on thebitangents of four circles tangent to a fifth circle was published byJohn Casey in 1866;^[5] Feuerbach's theorem has also been used as a test case forautomated theorem proving.^[6] The three points of tangency with the excircles form theFeuerbach triangle of the given triangle.

Theincircle of a triangleABC is acircle that is tangent to all three sides of the triangle. Its center, theincenter of the triangle, lies at the point where the three internal angle bisectors of the triangle cross each other.

Thenine-point circle is another circle defined from a triangle. It is so called because it passes through nine significant points of the triangle, among which the simplest to construct are themidpoints of the triangle's sides. The nine-point circle passes through these three midpoints; thus, it is thecircumcircle of themedial triangle.

These two circles meet in a single point, where they aretangent to each other. That point of tangency is the Feuerbach point of the triangle. In an equilateral triangle, the nine-point circle is the same as the incircle, and the Feuerbach point is therefore not defined.

Associated with the incircle of a triangle are three more circles, theexcircles. These are circles that are each tangent to the three lines through the triangle's sides. Each excircle touches one of these lines from the opposite side of the triangle, and is on the same side as the triangle for the other two lines. Like the incircle, the excircles are all tangent to the nine-point circle. Their points of tangency with the nine-point circle form a triangle, the Feuerbach triangle.

The Feuerbach point lies on the line through the centers of the two tangent circles that define it. These centers are theincenter andnine-point center of the triangle.^[1][2]

Let${\displaystyle x}$,${\displaystyle y}$, and${\displaystyle z}$ be the three distances of the Feuerbach point to the vertices of themedial triangle (the midpoints of the sidesBC=a, CA=b, andAB=c respectively of the original triangle). Then,^[7][8]

${\displaystyle x+y+z=2max(x,y,z),}$

or, equivalently, the largest of the three distances equals the sum of the other two. Specifically, we have${\displaystyle x=\frac{R}{2OI}|b-c|,y=\frac{R}{2OI}|c-a|,z=\frac{R}{2OI}|a-b|,}$ whereO is the reference triangle'scircumcenter andI is itsincenter.^{[8]: Propos. 3}

The latter property also holds for the tangency point of any of the excircles with the nine–point circle: the greatest distance from this tangency to one of the original triangle's side midpoints equals the sum of the distances to the other two side midpoints.^[8]

If the incircle of triangle ABC touches the sidesBC, CA, AB atX,Y, andZ respectively, and the midpoints of these sides are respectivelyP,Q, andR, then with Feuerbach pointF the trianglesFPX,FQY, andFRZ are similar to the trianglesAOI, BOI, COI respectively.^{[8]: Propos. 4}

Thetrilinear coordinates for the Feuerbach point are^[2]

${\displaystyle 1-\cos (B-C):1-\cos (C-A):1-\cos (A-B).}$

Itsbarycentric coordinates are^[8]

${\displaystyle (s-a)(b-c{)}^2:(s-b)(c-a{)}^2:(s-c)(a-b{)}^2,}$

wheres is the triangle'ssemiperimeter,${\displaystyle s=\frac{1}{2}(a+b+c)}$.

The three lines from the vertices of the original triangle through the corresponding vertices of the Feuerbach triangle meet at another triangle center, listed as X(12) in the Encyclopedia of Triangle Centers. Its trilinear coordinates are:^[2]

${\displaystyle 1+\cos (B-C):1+\cos (C-A):1+\cos (A-B).}$

• Thébault, Victor (1949), "On the Feuerbach points",American Mathematical Monthly,56 (8):546–547,doi:10.2307/2305531,JSTOR 2305531,MR 0033039.
• Emelyanov, Lev; Emelyanova, Tatiana (2001), "A note on the Feuerbach point",Forum Geometricorum,1: 121–124 (electronic),MR 1891524.
• Suceavă, Bogdan; Yiu, Paul (2006), "The Feuerbach point and Euler lines",Forum Geometricorum,6:191–197,MR 2282236.
• Vonk, Jan (2009), "The Feuerbach point and reflections of the Euler line",Forum Geometricorum,9:47–55,MR 2534378.
• Nguyen, Minh Ha; Nguyen, Pham Dat (2012), "Synthetic proofs of two theorems related to the Feuerbach point",Forum Geometricorum,12:39–46,MR 2955643.

• Triangle centers
• Theorems about triangles and circles

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