J. geom. 75 (2002) 106 – 1120047–2468/02/020106 – 07© Birkhauser Verlag, Basel, 2002DOI 10.1007/s00022-002-1490-0

The Gergonne and Nagel centers of a tetrahedron

Mowaffaq Hajja and Peter Walker

Abstract. The Gergonne center of a triangle is the intersection of the cevians through the points where the incircletouches the sides. This does not admit a direct generalization to tetrahedra since the cevians of a tetrahedronthrough the points where the insphere touches the faces are not necessarily concurrent. This article introduces analternative definition of the Gergonne center that coincides with the previous definition for the triangle and thatadmits a generalization to tetrahedra. The same is done for the Nagel center.

Mathematics Subject Classification (2000): 51.Key words: Gergonne center, Nagel center.

1. Introduction

The Gergonne center [4] of a triangle ABC is defined to be the point of intersection of thecevians AA′, BB ′, CC′ where A′, B ′, C′ are the points where the incircle of ABC touchesthe sides BC, CA, AB (Figure 1). Such cevians are concurrent by Ceva’s Theorem sinceA′, B ′, C′ can be equivalently defined by the algebraic relations

AB ′ = AC′, BC′ = BA′, CA′ = CB ′ (1)

The Nagel center is defined similarly by taking A′, B ′, C′ to be the points where the excirclestouch the sides (Figure 1). It can be easily seen that the points A′, B ′, C′ can also be definedby the relations

BC′ = CB ′, CA′ = AC′, AB ′ = BA′ (2)

This note is a result of the question whether Gergonne and Nagel centers can be defined fora general tetrahedron in a similar way.

Before answering this question (in Theorems 1 and 2), we remark that while the incenter,circumcenter, the centroid and the Fermat-Torricelli center of a general tetrahedron can bedefined in exactly the same way as they are defined for a triangle, the same cannot be said ofthe orthocenter. In fact, tetrahedra with an orthocenter (i.e. with concurrent altitudes) are

106

Vol. 75, 2002 The Gergonne and Nagel centers of a tetrahedron 107

A′

B′C′

CB

A

A′ CB

B′C′

A

Figure 1

characterized by the rather restrictive property that their nonadjacent edges are orthogonal[1, Theorem 212, p. 71 and Exercise 3, p. 73]. This should remove the impression, if any,that all results concerning the centers of a triangle have analogues for a tetrahedron.

2. The Gergonne center

It is most natural to define the Gergonne center of a tetrahedron ABCD to be the pointof intersection (if any) of the cevians AA′, BB ′, CC′, DD′ where A′, B ′, C′, D′ are thepoints where the insphere touches the faces. As seen in [3] and [2], for example, suchcevians are concurrent only for a limited class of tetrahedra. Thus we turn our attention tothe possibility of an alternative characterization of A′, B ′, C′, D′ parallel to the one givenby (1).

To this end, we note that (1) defines the points A′, B ′, C′ in terms of the six segmentsinto which these points divide the sides. The defining property is the requirement thatany two segments that share a vertex are equal. For a tetrahedron ABCD, one may thenconsider defining A′, B ′, C′, D′ in terms of the 12 triangles into which they divide thefaces. In analogy with the case of a triangle, one may suggest as the defining property therequirement that any two of the 12 resulting triangles that share a same edge have equalareas. This condition is shown in Figure 2 where our tetrahedron is opened flat and whereequal areas are designated by the same letter. Under this condition and the requirement thatthe cevians AA′, BB ′, CC′, DD′ are concurrent, we will see that the points A′, B ′, C′, D′exist and are unique. The point of intersection of AA′, BB ′, CC′, DD′ will be called theGergonne center.

The generalized Ceva’s Theorem [6] states that the cevians AA′, BB ′, CC′, DD′ areconcurrent if and only if one can assign masses MA, MB , MC , MD to the vertices in such a

108 Mowaffaq Hajja and Peter Walker J. Geom.

X

zz yy

x

x

Y Z

Z

D

A

DC D

B

Y

X

Figure 2

way that the center of mass of each three of them is one of the points A′, B ′, C′, D′. FromFigure 2, this is equivalent to the conditions

MA : MB : MC = x : y : z , MB : MC : MD = Z : Y : x ,

MC : MD : MA = X : y : Z , MD : MA : MB = z : Y : X.

These in turn can be seen to be equivalent to the conditions

Xx = Yy = Zz (= t, say) (3)

Denoting the areas of the faces by α, β, γ, δ, we see that

x + y + z = δ (4)

x + Y + Z = α (5)

X + y + Z = β (6)

X + Y + z = γ (7)

Adding (5), (6) and (7) and subtracting (4), we get

X + Y + Z = (α + β + γ − δ)/2 (8)

Subtracting (8) from (5) and using the fact that X = t/x, we obtain

x − t/x + (−α + β + γ − δ)/2 = 0

and hence

x2 + x(−α + β + γ − δ)/2 − t = 0.

Vol. 75, 2002 The Gergonne and Nagel centers of a tetrahedron 109

Doing the same for y and z and setting

a = (−α + β + γ − δ)/2 , b = (α − β + γ − δ)/2 ,

c = (α + β − γ − δ)/2 (9)

we see that

x2 + ax − t = 0, y2 + by − t = 0, z2 + cz − t = 0 (10)

Assuming that x, y, z are > 0, it follows that

2x = −a +√

a2 + 4t , 2y = −b +√

b2 + 4t , 2z = −c +√

c2 + 4t (11)

Adding these equations and using (4), it follows that

2δ = − (a + b + c) + (√

a2 + 4t +√

b2 + 4t +√

c2 + 4t) (12)

From (9) it follows that

2δ + (a + b + c) = (α + β + γ + δ)/2 (13)

This with (12) imply that

√a2 + 4t +

√b2 + 4t +

√c2 + 4t = (α + β + γ + δ)/2 (14)

As a function of t ∈ [0, ∞),√

a2 + 4t + √b2 + 4t + √

c2 + 4t is increasing and its rangeis [| a | + | b | + | c |, ∞). In view of (9), the existence of a (unique) t > 0 that satisfies(14) is therefore equivalent to the condition that

α + β + γ + δ

2≥

∣∣∣∣−α + β + γ − δ

2

∣∣∣∣ +∣∣∣∣α − β + γ − δ

2

∣∣∣∣ +∣∣∣∣α + β − γ − δ

2

∣∣∣∣

To prove that this is satisfied for all tetrahedra, we consider the expression

α + β + γ + δ

2+ ε1

−α + β + γ − δ

2+ ε2

α − β + γ − δ

2+ ε3

α + β − γ − δ

2

and we let ε1, ε2, ε3 assume the values ±1. The eight quantities

α, β, γ, δ, α + β + γ − δ, β + γ + δ − α, γ + δ + α − β, δ + α + β − γ

that we obtain are positive since the sum of the areas of any 3 faces of a tetrahedron isgreater than the area of the fourth [5, Theorem 2].

Thus for every tetrahedron, there exists a unique value t0 > 0 of t that satisfies (14). Using(11) and then (3) we obtain the corresponding solutions for x, y, z, X, Y, Z. Conversely, itis not difficult to see that the solutions for x, y, z, X, Y, Z thus obtained do indeed satisfy(4), (5), (6) and (7). We summarize this in the following theorem:

110 Mowaffaq Hajja and Peter Walker J. Geom.

THEOREM 1. Let ABCD be any tetrahedron. Then there exists a unique M in the interiorof ABCD such that the 12 triangles into which the cevians through M divide the faces havethe property that any two that share an edge have equal areas. This M is called the Gergonnecenter of ABCD.

3. The Nagel center

Here again, instead of defining the Nagel center of the tetrahedron ABCD to be the inter-section of the cevians AA′, BB ′, CC′, DD′ where A′, B ′, C′, D′ are the points where theexspheres touch the faces, we define A′, B ′, C′, D′ in terms of the 12 triangles into whichthey divide the faces. In the case of a triangle, note that (2) defines A′, B ′, C′ in terms of thesix segments into which the points A′, B ′, C′ divide the sides by the requirement that anytwo segments that lie on two different sides and that share an endpoint each with the thirdside are equal. For a tetrahedron ABCD, one may then consider the requirement that anythree of the 12 resulting triangles that lie on three different faces and that share an edge eachwith the fourth face have equal areas. This is expressed in Figure 3 by designating equalareas by the same letter. Note here that the concurrency of AA′, BB ′, CC′, DD′ is alreadyguaranteed (by simply taking (MA, MB, MC, MD)=(x, y, z, t)). Letting α, β, γ, δ denotethe areas of the faces as before, we get the following equations:

x + y + z = δ (15)

t + y + z = α (16)

x + t + z = β (17)

x + y + t = γ (18)

These lead immediately to the solution

t = (−2α + β + γ + δ)/3 , x = (α − 2β + γ + δ)/3

y = (α + β − 2γ + δ)/3 , z = (α + β + γ − 2δ)/3.

This shows that an interior Nagel center (as defined here) exists only if the sum of theareas of any three faces of the tetrahedron is greater than twice the area of the fourth. Wesummarize this in the following theorem:

THEOREM 2. For a tetrahedron ABCD, the following two conditions are equivalent:

(i) There exists a unique M in the interior of ABCD such that the 12 triangles intowhich the cevians through M divide the faces have the property that any three thatlie on different faces and that share an edge each with the fourth face have equalareas.

Vol. 75, 2002 The Gergonne and Nagel centers of a tetrahedron 111

z

zt ty

x

t

x x

y

D

A

DC D

B

z

y

Figure 3

(ii) The sum of the areas of any three faces of the tetrahedron is greater than twice thearea of the fourth.

When this point M exists, we call it the Nagel center of ABCD.

REMARKS. The referee kindly drew our attention to related work done by the lateVictor Thebault. In [7], Thebault defines the Nagel center of a tetrahedron by require-ments equivalent to (15)–(18) above. Thus that definition of the Nagel center coincideswith ours. In [8], the Gergonne center of a tetrahedron A1A2A3A4 is defined to be the pointwhose barycentric coordinates are

tan α1 : tan α2 : tan α3 : tan α4

where αj is the angle that the bisector of the trihedral angle at Aj makes with each of thefaces that contain Aj . Beside being the algebraic analogue of one of the characterizationsof the Gergonne center of a triangle, this definition is motivated by geometric consider-ations detailed there. It would be interesting to characterize those tetrahedra for whichThebault’s definition of the Gergonne center coincides with ours. Finally, we remark thatgeneralizations to higher dimensions are left for future projects.

Acknowledgement.

We are grateful to the referee for his valuable comments and for drawing our attentionto References [7] and [8] and to Professor Roland Eddy for drawing our attention toReference [2]. We would also like to thank Ayman Hajja for drawing the figures.

112 Mowaffaq Hajja and Peter Walker J. Geom.

References

[1] Altshiller-Court, N., Modern Pure Solid Geometry, Second Edition, Chelsea, NY 1964.[2] Fritsch, R., Tetraeder und Kugeln, Shaker Verlag, Aachen 2000.[3] Hajja, M. and Walker, P., The inspherical Gergonne center of a tetrahedron, preprint.[4] Kimberling, C., Central points and central lines in the plane of the triangle, Math. Mag. 67 (1994), 163–187.[5] Klamkin, M. S., Vector proofs in solid geometry, Amer. Math. Monthly 78 (1970), 1051–1065.[6] Landy, S., A generalization of Ceva’s theorem to higher dimensions, Amer. Math. Monthly 95 (1988),

936–938.[7] Thebault, V., Nagel point in the tetrahedron, Amer. Math. Monthly 54 (1947), 275–276.[8] Thebault, V., Sur des points de Gergonne et de Nagel d’un tetraedre, Math. Gaz. 33 (1949), 270–272.

Mowaffaq Hajja Peter WalkerMathematics American University of SharjahYarmouk University P. O. Box 26666–SharjahIrbid-Jordan United Arab Emiratese-mail: mhajja@yu.edu.jo e-mail: peterw@aus.ac.ae

Received 4 February 2000; revised 1 May 2000.