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Coaxal Pencil of Circles and Spheres in the Pavillet Tetrahedron

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Coaxal Pencil of Circles and Spheres in the Pavillet Tetrahedron 17TH INTERNATIONAL CONFERENCE ON GEOMETRY AND GRAPHICS ©2016 ISGG 4–8 AUGUST, 2016, BEIJING, CHINA COAXAL PENCILS OF CIRCLES AND SPHERES IN THE PAVILLET TETRAHEDRON Axel PAVILLET [email protected] ABSTRACT: After a brief review of the properties of the Pavillet tetrahedron, we recall a theorem about the trace of a coaxal pencil of spheres on a plane. Then we use this theorem to show a re- markable correspondence between the circles of the base and those of the upper triangle of a Pavillet tetrahedron. We also give new proofs and new point of view of some properties of the Bevan point of a triangle using solid triangle geometry. Keywords: Tetrahedron, orthocentric, coaxal pencil, circles, spheres polar 2 Known properties We first recall the notations and properties (with 1 Introduction. their reference) we will use in this paper (Fig.1). • The triangle ABC is called the base trian- gle and defines the base plane (horizontal). The orthocentric tetrahedron of a scalene tri- angle [12], named the Pavillet tetrahedron by • The incenter, I, is called the apex of the Richard Guy [8, Ch. 5] and Gunther Weiss [16], tetrahedron. is formed by drawing from the vertices A, B and 0 0 0 C of a triangle ABC, on an horizontal plane, • The other three vertices (A ,B ,C ) form a three vertical segments AA0 = AM = AL = x, triangle called the upper triangle and define BB0 = BK = BM = y, CC0 = CL = CK = z, a plane called the upper plane. where KLM is the contact triangle of ABC. We As a standard notation, all points lying on the denote I the incenter of ABC, r its in-radius, base plane will have (as much as possible) a 0 0 0 and consider the tetrahedron IA B C . standard triangle geometry notation, (e.g. O, cir- C0 cumcenter of the base triangle); all points lying on the upper plane will be denoted with primes (e.g. vertices A0,B0,C0, circumcenter O0, . ); all internal points of the tetrahedron will be de- noted with double primes (e.g. orthocenter H00, 00 C circumcenter O , . ). The three circles CA, CB, B0 and C centered at A, B and C with radius x, 0 C A z y and z will be called the three mutually tan- L K gent circles (dashed circles, externally tangent, I y A x M in Fig.1). B This tetrahedron attached to the base triangle This figure is rich in properties and various of (vertices A,B,C) is unique. It is unique for two them have been described in [12], [10] and [13]. reasons, firstly this is an asymmetric tetrahedron, Paper #010 the apex I and the vertices A0,B0,C0 have differ- r. It is called the incircle sphere [12, ent properties (most of the types of tetrahedron Theorem 5.2]. studied in the literature have symmetric proper- – An imaginary sphere centered at H00 ties for all the vertices and faces); secondly, there √ with radius n 2i where n = H00G is a one-to-one correspondence between the sca- e [12, § 6]. lene base triangle and the always acute (cf. [12, Theorem 5.3]) upper triangle and the tetrahedron • We define the circumcircle sphere of the linking them is unique. The study of this tetra- base triangle as we defined the incircle hedron links solid and triangle geometry, it of- sphere, i.e. the sphere which has the cir- fers an edge view of triangle geometry. It can cumcircle of the base triangle as great cir- be used two ways, using solid geometry to prove cle. In [10, Corollary 3.1], we proved that triangle geometry properties and theorems or us- the trace on the upper plane of the circum- ing triangle geometry to prove properties of the circle sphere is the Euler circle of the up- tetrahedron. per triangle; a quicker proof is given in [13, Theorem 10]. • The tetrahedron IA0B0C0 is orthocentric (because IB02 = 2y2 + r2, while A0C02 = On figure3, is drawn the circumcircle, 2x2 + 2z2 so that the sum IB02 + A0C02 is the trace of the circumcircle sphere on the independent of the sides) [12, p. 1]. plane defined by the line OI and the fourth altitude; this plane, called the Euler line • The lines A0K, B0L and C0M are three alti- plane, is perpendicular to the upper plane tudes of the tetrahedron [12, Theorem 1.2]. and intersects it along the Euler line of the • The altitude of the tetrahedron going upper triangle [10, Corollary 3.2 , (a) and through its apex I is called the fourth al- (b)]. The Euler circle of the upper triangle titude. is centered at ω0. • The orthogonal projection of the orthocen- We describe the coaxal pencils of circles and ter H00 of the tetrahedron on the plane of spheres we are going to use, noting that by ap- ABC is Ge, the Gergonne point of ABC plication of the always acute property (cf. [12, [12, Theorem 1.3]. Theorem 5.3]), these pencils are only of the non- intersecting type. We start with the most well- We will also use: known. • the five mutually orthogonal polar spheres of the orthocentric group A0,B0,C0,I,H00. Represented on fig.2 by their traces on the 3 The Euler pencil of an or- base plane, on the three vertical planes de- fined by AA0BB0 and on their siblings, they thocentric tetrahedron are: As described in [1], any orthocentric tetrahe- – The sphere centered at A0 with ra- √ dron, generates a coaxal pencil of spheres which dius x 2. It intersects the base plane axis is the Euler line of the tetrahedron and con- along the corresponding mutually tan- tains the circumsphere (centered at O00), the first gent circle. This sphere and its two twelve point sphere (centered at the centroid siblings are called vertex spheres [12, G00), the second twelve point spheres (center de- Theorem 5.2]. noted Γ00 on fig.4) and the polar sphere of the – The sphere centered at I with radius tetrahedron (cf. [1, § 805]). The orthocentroidal 2 C0 A0 H0 00 C 0 H L A B Ge I M K B Figure 1: Orthocenter and Altitudes. C0 M0 C H0 A0 H00 L Ge K B0 I A M Fl B L0 Figure 2: The Polar Spheres of the Orthocentric Group 3 K0 C0 0 0 B Am m O0 G0 C ω0 H0 0 A0 Cm 00 H B0 O L K I G A e M B Figure 3: Trace of the Circumcircle Sphere on the Upper Plane sphere, the sphere having the segment H00G00 the pencil of spheres along a line which is joining the orthocenter to the centroid as diam- the radical axis of the system of circles. eter, also belongs to this pencil (cf. [1, § 807]). • The line of centers of the coaxal pencil Finally the orthic plane of the tetrahedron (per- 00 of circles is the orthogonal projection of pendicular to the Euler line at H on fig.4) is r the line of centers of the pencil of spheres the radical plane of this coaxal pencil (cf. [1, upon the secant plane and therefore they are § 809]). We call this coaxal pencil the Euler pen- coplanar. cil of an orthocentric tetrahedron. Remark 4.1 A special case arises when the plane of intersection is the radical plane of the 4 Intersection of a coaxal pen- pencil because in this case we only get the basic circle, real or imaginary, of the pencil (provided cil of spheres with a plane we do not consider the radical plane as a sphere of the pencil). It is clear that the intersection of a coaxal pencil of quadric with a plane yields a coaxal pencil of conic [14, § 13.15]. 5 The Euler pencil of a trian- When we deal with spheres, a theorem by Court gle. states [1, § 558]: Theorem 4.1 Any plane cuts a coaxal pencil of From Theorem 4.1, the intersection of the Euler spheres along a coaxal pencil of circles. pencil of an orthocentric tetrahedron with one of its face therefore yields a coaxal pencil of cir- cles. Now, the orthogonal projection of the Euler It comes with the following properties: line of an orthocentric tetrahedron on one of its • The secant plane cuts the radical plane of face is the Euler line of the face triangle: indeed 4 C0 00 0 O 0 Bm Am O0 Q G0 ω0 H0 00 m Γ0 G Γ00 H0 0 00 Cm Ht A0 B0 H00 P 0 00 I Ho Hr 00 Hs 0 Hp Figure 4: Cross-section of the Euler pencil of an orthocentric tetrahedron along both the Euler line plane and the upper plane. orthocenter and circumcenter of the tetrahedron and one given by the intersection with the first become orthocenter and circumcenter of the face twelve point sphere and the Euler circle. Hence triangle by orthogonal projections. Moreover the the pencils of spheres and the pencil of circles trace of the circumsphere on the face is the cir- determine the same involution on the Euler line cumcircle of the face triangle while the trace of and therefore the pencil of circles determined by the first twelve point sphere is its Euler circle. the trace of the pencil of spheres and the pencil determined by the circumcircle and Euler circle But we also know that, in a triangle, it exists a are identical. coaxal pencil of circles defined by the circumcir- cle and the Euler circle and from Desargue’s the- We call it the Euler pencil of the triangle (in our orem a coaxal pencil of quartics surfaces or con- case, the Euler pencil of the upper triangle).
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