Intriguing Invariants of Centers of Ellipse-Inscribed Triangles
aa r X i v : . [ m a t h . M G ] D ec INTRIGUING INVARIANTS OF CENTERSOF ELLIPSE-INSCRIBED TRIANGLES
MARK HELMAN, RONALDO GARCIA, AND DAN REZNIK
Abstract.
We describe invariants of centers of ellipse-inscribed triangle fam-ilies with two vertices fixed to the ellipse boundary and a third one whichsweeps it. We prove that: (i) if a triangle center is a fixed linear combinationof barycenter and orthocenter, its locus is an ellipse; (ii) and that over thefamily of said linear combinations, the centers of said loci sweep a line; (iii)over the family of parallel fixed vertices, said loci rigidly translate along a sec-ond line. Additionally, we study invariants of the envelope of elliptic loci overcombinations of two fixed vertices on the ellipse.
Keywords
Ellipse, Locus, Invariant, Envelope, Lima¸con.
MSC Introduction
We describe a few intriguing invariants displayed by triangles centers [12] ofellipse-inscribed triangle families T ( t ) = V V P ( t ): V , V are fixed on the boundaryof an ellipse E , and P ( t ) sweeps said boundary. One readily observes that the locusof classic triangle centers such as the bary-, circum-, and orthocenter (denoted X k , k = 2 , ,
4, after [12]) are distinct ellipses; see Figure 1. Our main results are: • If a triangle center X k is a fixed linear combination of bary- X and or-thocenter X , its locus is an ellipse (co-discovered with A. Akopyan [1]).Experimentally, the reverse seems also true, though we do not prove it. • Over the continuous family of said linear combinations, centers of said locisweep a straight line. • Over the family of parallel V V , the elliptic locus of X k is a family ofellipses which rigidly translate along a line passing through O , the centerof E .Additionally, we study the family of elliptic loci of X k with V fixed over all V on E . We show that (i) the locus of their centers is an ellipse axis-aligned with E , andthat (ii) the external envelope to the locus family is invariant over V . In particular,the external envelope of X (resp. X ’s) loci is an affine image of Pascal’s Lima¸con(a 1/3-sized copy of E ). Related Work.
In [17], an explicit derivation is given for the elliptic locus of theorthocenter for the same triangle family studied in this article. Dykstra et al. [3]derived equations for loci of triangle centers under triangle families with two verticesaffixed to special points of the ellipse. Loci of triangle centers have been studiedfor alternative triangle families including: (i) Poristic triangles (those with fixed
Date : September, 2020.
Figure 1.
A triangle V V P ( t ) (blue) is inscribed in an ellipse E (black) with semi-axes a, b . While V , V are fixed, P ( t ) executes one revolution along the boundary. Consider triangle centers X k , k =2 , , L e (dashed green). For any choice of V V , the locus of (i) X is an axis-aligned ellipse (brown) with semi-axes a/ , b/
3; (ii) X is a segment (red) contained in the perp.bisector of V V ; (iii) X is an axis-aligned ellipse passing through V , V , with aspect ratio b/a .An intermediate point X ρ ( ρ = 1 /
2) is shown on L e as well as its elliptic locus (green) centeredon O ρ . Though an ellipse for any ρ , this locus is in general neither axis-aligned (notice its slantedaxes, dashed green), nor similar to E . Video incircle and circumcircle) [5, 29, 30, 19]. Odehnal describes pointwise, circular, andelliptic loci for dozens of triangle centers in the poristic family [20]. The poristicfamily is related via a similarity transform to another “famous” family: 3-periodicsin the elliptic billiard [8]; (ii) triangles with common incircle and centroid. Pamfiloshas shown their vertices lie on a conic [21]; (iii) Triangles with sides tangent to acircle [14]; (iv) Triangles associated with two lines and a point not on them [27],etc. (v) Stanev has desrives the (ellptic) locus for the centroid of ellipse-inscribedequilaterals [28].Some examples of polygon families include (i) rectangles inscribed in smoothcurves [25], (ii) Poncelet polygons and the locus of their centroids [26] or theircircumcenter [2].We’ve studied loci of 3-periodics in the elliptic billiard, having found experimen-tally that the locus of the incenter is an ellipse [22]. This was subsequently proved[24]. Appearing thereafter were proofs for the elliptic locus of the barycenter [15]and circumcenter [4, 6]. More recently, with the help of a computer algebra system(CAS), we showed that 29 out of the first 100 entries in [12] are ellipses over billiard3-periodics, though what determines ellipticity is still not understood [9]. We’ve NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 3
Figure 2.
Relative locations of the first 16 triangle centers on [12] which are both on the Eulerline and which are fixed linear combinations of X and X . also studied properties (e.g., area invariance) of the negative pedal curve (NPC)of the ellipse with respect to a point on its boundary [10] as well as its pedal-likederivatives [23]. We have also studied the locus of Brocard points over circle- andellipse-inscribe triangle families [7].The envelope of Euler lines in triangles with two fixed vertices at the foci of anellipse was studied in [16]. Note: we study loci of this family in Appendix C. Article Structure.
Section 2 defines basic concepts used in the article. Section 3contains our main results. Section 4 analyzes loci over families of parallel V V .Section 5 analyzes the envelope of loci with fixed V and varying V . In the con-clusion, Section 6, a table is included with videos mentioned throughout the paper.A set of appendices is included: Appendix C overviews triangle centers whose lociare ellipses for a closely-related triangle family: V V are fixed on the foci of E .Appendix A includes longer calculations used in derivations of key properties ofelliptic loci. Finally, Appendix D contains a quick-reference to most symbols usedherein. 2. Definitions
Let ( a, b ) be the semi-axes of ellipse E centered on O . Let U ( s ) = [ a cos s, b sin s ]be a generic point on its boundary. Let t , t be two constants, and define V = U ( t ), V = U ( t ). Consider the triangle family T ( t ) = V V P ( t ), where P ( t ) = U ( t ), t ∈ [0 , π ); see Figure 1.Given a T ( t ), Let X ρ be a fixed linear combination of barycenter X and ortho-center X (both on Euler Line L e ), i.e., X ρ = (1 − ρ ) X + ρX ; Figure 2 depictsthe relative locations of the first 16 centers on [12] with fixed ρ , whereas Table 1specifies their respective ρ . X k
20 550 376 548 3 549 631 140 632
547 5 381 546 ρ -2 -1.25 -1 -0.875 -0.5 -0.25 -0.2 -0.125 -0.05 Table 1.
First 16 points on [12] on Euler line and at fixed ρ . Currently, upwards of 38k+ triangle centers are catalogued in [12]. These arepoints on the plane of a triangle whose trilinear (or barycentric) coordinates arefunctions of side lengths and angles. Said functions must satisfy homogeneity,bisymmetry, and cyclicity [31, Triangle Center]. Examples include such classiccenters as the incenter X , barycenter X , circumcenter X , orthocenter Remark . Since X ρ is a linear combination of two triangle centers, its barycentricsadd in a similar manner and therefore X ρ is also a triangle center [18].Finally, let c = a − b , d = a + b , and z = cos( t + t ). MARK HELMAN, RONALDO GARCIA, AND DAN REZNIK Main Results
Referring to Figure 1:
Remark . Since X = [ V + V + P ( t )] /
3, the locus of X is an ellipse E centeredon O = ( V + V ) /
3, axis-aligned with E , and with semi-axes ( a , b ) = ( a/ , b/ Proposition 1.
For any V , V , the locus of X is an ellipse E of semi-axes a , b ,centered on O , axis-aligned with E , and with aspect ratio b/a . These are given by: O = c (cid:20) cos t + cos t a , sin t + sin t b (cid:21) ( a , b ) = ( w/a, w/b ) , where: w = √ (cid:16)p a + b + ( b − a ) z (cid:17) Proof.
Using a CAS for simplification obtain the following parametric for E : E : [ x ( t ) , y ( t )] , t ∈ [0 , π ) , where: x ( t ) = d (cos t + cos t + cos t ) + c cos( t + t + t )2 ay ( t ) = d (sin t + sin t + sin t ) + c sin( t + t + t )2 b (cid:3) Theorem 1.
For any V , V , and any ρ , the locus of X ρ is an ellipse (in generalnot axis-aligned with E ) with parametric: E ρ : [ x ρ ( t ) , y ρ ( t )] , t ∈ [0 , π ) , where: x ρ = ( a ( ρ + 2) + 3 b ρ )(cos t + cos t + cos t )6 a − c ρ cos( t + t + t )2 ay ρ = ( a ( ρ + 2) + 3 b ρ )(sin t + sin t + sin t )6 b − c ρ sin( t + t + t )2 b and center O ρ at: O ρ = " (cid:0) a ( ρ + 2) + 3 b ρ (cid:1) (cos t + cos t )6 a , (cid:0) a ρ + b ( ρ + 2) (cid:1) (sin t + sin t )6 b Proof.
CAS manipulation of X ρ = (1 − ρ ) X + ρX . (cid:3) in Appendix B we’ve included all 226 triangle centers on [12] which are fixed linearcombinations of X , X .As before, let U ( s ) denote [ a cos s, b sin s ] on E . Lemma 1.
Lines U ( t ) U ( t ) are parallel for all t + t = t , where t is someconstant.Proof. The slope of the line through points U ( t ) = [ a cos( t ) , b sin( t )] and U ( t ) =[ a cos( t ) , b sin( t )] is given by b (sin t − sin t ) a (cos t − cos t ) = 2 b cos( t + t ) sin( t − t ) − a sin( t + t ) sin( t − t ) = − ba cot (cid:18) t (cid:19) NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 5 which only depends on t . (cid:3) Remark . Lemma 1 implies that if V V are vertical (resp. horizontal), then t + t = 2 kπ (resp. t + t = (2 k + 1) π ), where k is an integer. Corollary 1.
For V V vertical or horizontal, for all ρ , the locus of X ρ is axis-parallel with E .Proof. Follows from the implicit equation for X ρ (Appendix A). Specifically, thecoefficient a of xy vanishes whenever t + t is an integer multiple of π . (cid:3) Let ( a ρ , b ρ ) denote the semi-axes of the locus of X ρ . Proposition 2.
The ratio a ρ /b ρ is invariant over the family of parallel V V .Proof. Rewrite the ellipse in Theorem 1 implicitly, and using a CAS obtain theratio of eigenvalues of its Hessian, yielding the expression for a ρ /b ρ in Appendix A.Notice its non-constant terms only depend on the sum t + t which with Lemma 1yields the claim. (cid:3) Corollary 2.
For exactly three values of ρ , namely, , / , (corresponding to X , X , and X ), the aspect ratio of the corresponding elliptic locus is independent of V and V , i.e., it only depends on ( a, b ) .Proof. For ρ ∈ { , / } the aspect ratio is equal to a/b and for ρ = 1 / a + b ) / (2 ab ) . From Appendix A it follows that λ ρ = a ρ /b ρ is a function of[ ρ, cos( t + t )] = ( ρ, z ). The function λ ρ ( ρ, z ) is independent of z when the levelset ∂λ ρ /∂z = 0 is a vertical straight line. Direct analysis shows that is the caseexactly when ρ ∈ { , / , } . (cid:3) Proposition 3.
The product a ρ b ρ is invariant over the family of parallel V V andgiven by: a ρ b ρ = (cid:12)(cid:12) (2 ρ + 1)(2 a b ( ρ −
1) + 3 ρ ( a − b ) cos( t + t ) − ρ ( a + b )) (cid:12)(cid:12) ab Proof.
Obtain the above from Equation 1 in AppendixA. Since for parallel V V , t + t is constant (Lemma 1), the proof is complete. Alternatively, the same resultcould be obtained by symbolic simplification of the affine curvature of X ρ equal to( a ρ b ρ ) − / , since it is an ellipse [11]. (cid:3) Remark . Only for ρ ∈ { , − / } , i.e., X ρ ∈ { X , X } , is the product a ρ b ρ inde-pendent of V V , and equal to ab/ Corollary 3 (Axis annihilation) . For each family of parallel V V , there is a unique ρ (other than − / ) such that the product a ρ b ρ = 0 . Said ρ is given by: ρ = 2 a b a − b ) cos( t + t ) + 2 a b − a − b except when the denominator vanishes, which can only happen if a/b < √ . In thiscase, no ρ exists.Proof. Follows from the expressionm for a ρ b ρ in Proposition 3. (cid:3) Remark . It a/b = √
3, for vertical V V , the axis can’t be annihilated. MARK HELMAN, RONALDO GARCIA, AND DAN REZNIK
Figure 3.
The loci of X ρ (green) can only be circles when V V (blue) are horizontal (left) orvertical (right). When horizontal (resp. vertical) only at ρ = { ρ h, , ρ h, } (resp. ρ = { ρ v, , ρ v, } )is the locus a circle. As the V V traverse all horizontals (resp. verticals), the circles will rigidlytranslate vertically (resp. horizontally). Video Referring to Figure 4:
Corollary 4.
The semi-axis lengths a ρ , b ρ of the locus of X ρ are invariant over thefamily of parallel V V .Proof. By Proposition 2 and Lemma 3, the ratio a ρ /b ρ and product a ρ b ρ of theaxes are invariant over parallel V V , respectively. The result follows. (cid:3) Referring to Figure 3:
Proposition 4.
The locus of X ρ is a circle iff V V is (i) horizontal with ρ assumingtwo values ρ h , or (ii) vertical with a =3 b , and ρ assuming two values ρ v . These aregiven by: ρ h = bb ± a , ρ v = aa ± b Proof.
The parametrization for ( x ρ , y ρ ) in Theorem 1 can be developed to yield: • For t + t = π and ρ = b/ ( b + 3 a ), X ρ is a circle centered in [0 , ( a + b ) sin t / (3 a + b )] and radius a ( a + b ) / ( b + 3 a ). • For t + t = π and ρ = b/ ( b − a ), X ρ is a circle centered in [0 , ( a − b ) sin t / ( b − a )] and radius a ( a − b ) / (3 a − b ). • For t + t = 0 and ρ = a/ ( a + 3 b ), X ρ is a circle centered in [( a + b ) cos t / ( a + 3 b ) ,
0] and radius b ( a + b ) / ( a + 3 b ). • For t + t = 0, ρ = a/ ( a − b ) and a = 3 b , X ρ is a circle centered in[( a − b ) cos t / ( a − b ) ,
0] and radius | b ( a − b ) / ( a − b ) | .If V V is neither horizontal nor vertical, there are no real solutions for ρ suchthat a ρ /b ρ = 1. (cid:3) Proposition 5.
For V V vertical, consider the case of a = 3 b and ρ / ∈ {− / , / } .The locus of X ρ is the axis-aligned ellipse centered at [2 b (2 r + 3) cos t / , and axes a ρ = b | r − | / , b ρ = b | r + 1 | / . This ellipse is a circle when ρ = 1 / , i.e., when X ρ = X .Proof. Direct derivation from Theorem 1. (cid:3)
NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 7
Remark . By definition, X is contained in the perpendicular bisector of V V ,given by: − a sin( t + t ) x + 2 b (cos( t + t ) + 1) y + c (cos t + cos t ) sin( t + t ) = 0 Proposition 6.
The locus of X is a variable-length segment P P ′ given by: P ,x = c a (1 − cos(2 t + 2 t )) p − t + t ) + cos t + cos t ! P ,y = − c b (cid:16)p − t + t ) + sin t + sin t (cid:17) P ′ ,x = c a (cos(2 t + 2 t ) − p − t + t ) + cos t + cos t ! P ′ ,y = c b (cid:16)p − t + t ) − sin t − sin t (cid:17) Furthermore, its length L is given by: L = | P − P ′ | = c p d − c cos ( t + t ))2 ab Proof.
The coordinates of X = [ x , y ] are given by: x = c a (cos( t + t + t ) + cos t + cos t + cos t ) y = c b (sin( t + t + t ) − sin t − sin t − sin t )Direct calculations yield the claimed expressions. (cid:3) Corollary 5.
The min (resp. max) of L is c /a (resp. c /b ) and the midpoint of P P ′ is given by (cid:20) c a (cos t + cos t ) , − c b (sin t + sin t ) (cid:21) Proof.
Direct from the Proposition 6. (cid:3) Locus Center Translation
Referring to Figure 4:
Proposition 7.
Over the family of parallel V V , the locus of X ρ is a familyof rigidly-translating ellipses whose center moves along a straight line L k passingthrough O and given by: L k : y = ab · (3 a + b ) ρ + 2 b ( a + 3 b ) ρ + 2 a tan (cid:18) t + t (cid:19) x Proof.
Directly from the expression for O ρ in Theorem 1. (cid:3) Remark . L k is perpendicular to V V when ρ = 1 ( X ). MARK HELMAN, RONALDO GARCIA, AND DAN REZNIK
Figure 4.
Over the family of parallel V V (dashed blue, constant t + t ), the loci of X ρ (solidgreen) are a family of rigidly-translated ellipses. Their centers (green dots) move along a straightline L k (magenta) which crosses E ’s center. Shown are the cases for ρ ∈ {− , − / , , / } , i.e., X k , k = 376 , , ,
5, respectively. Video
Referring to Figure 5:
Proposition 8.
For V V stationary, as one varies ρ , the center O ρ of the locusof X ρ follows a straight line L ρ whose slope only depends on the slope of V V . Infact: L ρ : (cid:20) a (cos t + cos t )3 , b (sin t + sin t )3 (cid:21) ++ ρ cos (cid:18) t − t (cid:19) (cid:20) a + 3 b a cos (cid:18) t + t (cid:19) , a + b b sin (cid:18) t + t (cid:19)(cid:21) Proof.
Follows from Theorem 1. (cid:3)
Corollary 6.
The product of the slopes of L ρ and V V is constant over all choicesof V and V and equal to − a + b a +3 b . Corollary 7.
Only when V V coincides with either the major or minor axis of E can L ρ pass through the center of E . NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 9
Figure 5.
Elliptic loci (green ellipses) are shown for ρ ∈ { , / , / , } with centers O k , k =2 , , , L ρ (dashed orange). Also shown is a family of lines L k (magenta) passingthrough the center of E depicting the locus of individual O ρ as the family of parallel V V istraversed. Also shown (solid orange) is the tricuspid envelope of L ρ for fixed V , over all V on E .Video Referring to Figure 6 (bottom right):
Remark . When V V passes thru O , L ρ collapses to O , and O ρ = O for all ρ . Corollary 8. O ρ is on V V at the following ρ : ρ = 2 a b b − a ) cos( t + t ) + 2 a b + 3 a + 3 b Phenomena with V fixed and V variable Envelope of L ρ . Referring to Figure 5:
Proposition 9.
For fixed V , over all V on E , the envelope of L ρ is a deltoid-likecurve ∆ (with 3 cusps), given parametrically by: ∆ t ( u ) = " cos t (cid:0) a − b (cid:1) a (3 a + b ) , − sin t (cid:0) a − b (cid:1) b ( a + 3 b ) + " (2 cos u + cos ( t + 2 u )) (cid:0) a − b (cid:1) a ( 3 a + b ) , − (2 sin u − sin ( t + 2 u )) (cid:0) a − b (cid:1) b ( a + 3 b ) Figure 6.
Four positions of parallel V V are shown, approaching the center of E . For each position,loci for ρ = { , / , / , } are collinear on L ρ (dashed orange). Notice as V V approaches theposition where it crosses the origin, centers come closer to each other and L ρ degenerates to apoint (bottom right). Video Proof.
Follows from Proposition 8 and the definition of the envelope [11, Chapt.3]. (cid:3)
Proposition 10.
The area of the region bounded by ∆ t is invariant over t andgiven by A (∆ t ) = π ( a − b ) ab (3 a + b )( a + 3 b )5.2. Locus of O ρ . Referring to Figure 7:
Proposition 11.
With V fixed, over all V , the locus of centers O ρ of the loci of X ρ is an ellipse Γ ρ centered on O ′ ρ , which is axis-aligned with E and contains itscenter O . Its semiaxes ( a ′ , b ′ ) and center are given by: ( a ′ , b ′ ) = (cid:18) a ( ρ + 2) + 3 b ρ a , a ρ + b ( ρ + 2)6 b (cid:19) O ′ ρ = [ a ′ cos t , b ′ sin t ] NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 11
Figure 7.
For a choice of V and over V on E (black), the loci of X ρ are a family of ellipses (green)whose centers (green dots) sweep an ellipse Γ ρ passing through the center of E and centered O ′ ρ .Shown are the cases for ρ ∈ {− , − / , , / } , i.e., X k , k = 376 , , ,
5, respectively. Also shownare envelopes (pink) to the ellipse families. For X (top right), the envelope is a half-sized Steiner’shat [10]; for X (bottom left) the envelope is an ellipse of fixed axes, internally tangent to E atone point. All envelopes are area-invariant wrt V . Corollary 9. At ρ = 1 ( X ), Γ ρ is an axis-aligned ellipse with aspect ratio b/a with center at h ( a + b ) cos t a , ( a + b ) sin t b i and axes ( a + b a , a + b b ) . Corollary 10. At ρ = 0 ( X ), Γ ρ is an ellipse with aspect ratio a/b centered at (cid:2) a cos t , b sin t (cid:3) with axes ( a , b ) . Corollary 11.
Over all V the locus of O ′ ρ is an ellipse Γ ′ ρ whichi s axis-alignedand concentric with E . The semi-axes of Γ ′ ρ are also ( a ′ , b ′ ) . Referring to Figure 8:
Figure 8.
Locus Γ ρ (red) of the centers of X ρ (green) of X ρ for ρ = 2 /
3, when V coincides withthe left (resp. top) vertex of E (black). Notice Γ ρ (red) is tangent at the center of O to the minor(resp. major) axis of E and its center O ′ ρ lies on the major (resp. minor) axis of E . Remark . If V is fixed at the left (resp. top) vertex of E , over all V , Γ ρ isaxis-aligned with E and tangent at O to its minor (resp. major) axis.5.3. Envelope of the family of elliptic X ρ .Proposition 12. With V stationary and V sweeping the boundary of E , a regularpart of the envelope of X ρ is a curve Γ t parametrized by x t = (cid:2) ( a + 3 b )(2 cos t + cos t ) − c cos( t + 2 t ) (cid:3) ρ a + a t + cos t ) y t = (cid:2) (3 a + b )(2 sin t + sin t ) − c sin( t + 2 t ) (cid:3) ρ b + b t + sin t ) Proof.
Direct from the definition of an envelope [11] via CAS simplification. (cid:3)
Proposition 13.
The area (algebraic) of the region bounded by Γ t is invariantover t and given by A (Γ t ) = π " (cid:0) a + 2 a b + 15 b (cid:1) ρ ab + 2 (cid:0) a + 2 a b + 3 b (cid:1) ρab + 4 ab Proof.
Follows by direct integration of A (Γ t ) = R Γ t ( xdy − ydx ). (cid:3) In [10] we called
Steiner’s Hat the negative pedal curve of an ellipse with respectto a point M on the boundary. One of its curious properties is that it is area-invariant over all M . NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 13
Figure 9.
Let V be a point on E (black, rotated 90 degrees to save space, a/b = 2). Over V ,the loci of X are ellipses (green) which all pass through V . Their centers (green dots) sweep anaxis-aligned ellipse (red) centered at O ′ ρ , whose aspect ratio is b/a . The exterior envelope of saidloci is (in general) a non-convex curve (pink) which is the affine image of Pascal’s Lima¸con. Itsarea is independent of V , and it is tangent to E at at least one point. Video Remark . With V stationary and V sweeping the boundary of E , the envelopeof the locus of X over all positions of V is 2:1 homothetic to Steiner’s Hat.This stems from the fact that it cuts V V perpendicularly and at its midpoint.Referring to Figure 7 (bottom left) and Figure 9: Remark . For ρ = 0 (resp. ρ = 1), the external envelope of X ρ is internally(resp. externally) tangent to E at V (resp. at the point(s) on E whose normal goesthrough V ). For ρ = 0 the envelope is elliptic with axes 2 a/ b/ . Proposition 14.
For ρ = 1 , the external envelope of X ρ is the affine image ofPascal’s Lima¸con.Proof. A construction for Pascal’s Lima¸con is given in [31, Pascal’s Lima¸con] asfollows: specify a fixed point P and a circle C . Then draw all circles with centerson C which pass through P . The external envelope of said circles is the Lima¸con.Referring to Figure 9, apply an affine transformation that sends Γ ρ to a circle C .This will automatically send all X ρ ellipses to (variable radius) circles, since theyhave the same aspect ratio and are axis-aligned with Γ ρ (Proposition 11. Therefore,the affine image of the X ρ becomes a family of circles with centers on C through acommon point P , the affine image of V . (cid:3) Conclusion
This article studied properties of the loci of triangle centers over of a special ofellipse-inscribed triangles. The following questions are still unanswered: • Is there a triangle center which is not a fixed linear combination of X , X whose locus over T ( t ) is an ellipse? We did not find one amongst all 38k+centers listed in [12]. • For V fixed and over all V on E , are there interesting properties of theinternal envelope of the family of X ρ ellipses? For X , X , this envelope isa point ( O + ( V − O ) / V , respectively. However for other ρ thisenvelope is more complex. Animations illustrating the dynamic geometry of some of the above phenomenaappear on Table 2.Id Title youtu.be/...
01 Basic elliptic loci zjiNgfndBWg X ρ slides on Euler line w5KuN 0rQBQ
03 Locus of X ρ over parallel V V zFOeENDJRho
04 Relative motion of loci over parallel V V TpBjKlkFjkg
05 Circular loci if V V are horiz. or vert. nLeKvxcicNY
06 Lima¸con-Like envelopes of X locus family sPQrz7ddRfA Table 2.
Illustrative animations, click on the link to view it on YouTube and/or enter youtu.be/ as a URL in your browser, where
is the provided string.
We are very grateful to A. Akopyan and P. Moses for key insights. We thank P.N.de Souza for his crucial editorial help. The second author is fellow of CNPq andcoordinator of Project PRONEX/ CNPq/ FAPEG 2017 10 26 7000 508.
Appendix A. Axis ratio of locus of X ρ Here we assume the origin is at the center of the X ρ locus. Then the latter canbe expressed as: X ρ : a x + 2 a x y + a y + a = 0The aspect ratio is given by: a ρ b ρ = a + a + p ( a − a ) + 4 a a a − a )The product of axes is given by:(1) a ρ b ρ = | a | p a a − a where (recall z = cos( t + t )): a = 54 a ρc (cid:0) a ρ + b ρ + 2 b (cid:1) z − a (cid:0) a − b a + 5 b (cid:1) ρ − a b (cid:0) a + b (cid:1) ρ − a b a = 54 ρ ( ρ − abc p − z a = 54 b ρc (cid:0) a ρ + 3 b ρ + 2 a (cid:1) z − b (cid:0) a − b a + 9 b (cid:1) ρ − b (cid:0) a + 3 b (cid:1) a ρ − b a a = (2 ρ + 1) (cid:0) a ρz − b ρz − a ρ + 2 a b ρ − b ρ − b a (cid:1) NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 15
Appendix B. Triangle Centers at fixed ρ Amongst the 4.9k triangle centers on the Euler line [13], only the following 226are fixed linear combinations of X and X : X k , k =
2, 3, 4, 5, 20, 140, 376, 381, 382,546, 547, 548, 549, 550, 631, 632, 1564, 1656, 1657, 2041, 2042, 2043, 2044, 2045, 2046,2675, 2676, 3090, 3091, 3146, 3522, 3523, 3524, 3525, 3526, 3528, 3529, 3530, 3533, 3534,3543, 3545, 3627, 3628, 3830, 3832, 3839, 3843, 3845, 3850, 3851, 3853, 3854, 3855, 3856,3857, 3858, 3859, 3860, 3861, 5054, 5055, 5056, 5059, 5066, 5067, 5068, 5070, 5071, 5072,5073, 5076, 5079, 7486, 8703, 10109, 10124, 10299, 10303, 10304, 11001, 11539, 11540,11541, 11737, 11812, 12100, 12101, 12102, 12103, 12108, 12811, 12812, 14093, 14269,14782, 14783, 14784, 14785, 14813, 14814, 14869, 14890, 14891, 14892, 14893, 15022,15640, 15681, 15682, 15683, 15684, 15685, 15686, 15687, 15688, 15689, 15690, 15691,15692, 15693, 15694, 15695, 15696, 15697, 15698, 15699, 15700, 15701, 15702, 15703,15704, 15705, 15706, 15707, 15708, 15709, 15710, 15711, 15712, 15713, 15714, 15715,15716, 15717, 15718, 15719, 15720, 15721, 15722, 15723, 15759, 15764, 15765, 16239,16249, 16250, 16446, 17504, 17538, 17578, 17800, 18585, 18586, 18587, 19708, 19709,19710, 19711, 21734, 21735, 23046, 33699, 33703, 33923, 34200, 34551, 34552, 34559,34562, 35018, 35381, 35382, 35384, 35400, 35401, 35402, 35403, 35404, 35405, 35406,35407, 35408, 35409, 35410, 35411, 35412, 35413, 35414, 35415, 35416, 35417, 35418,35419, 35420, 35421, 35434, 35435, 35732, 35734, 35735, 35736, 35737, 35738, 36436,36437, 36438, 36439, 36445, 36448, 36454, 36455, 36456, 36457, 36463, 36466 . Appendix C. Focus-Mounted Triangles
Consider the 1d family of triangles T f ( t ) = f f P ( t ) where P ( t ) sweeps E andthe f j are the foci of E , at [ ± c, X , X were proved in [3, Thm 2.2.1]: Proposition 15.
Over the first 1000 triangle centers listed in [12] , only the loci of X k , k = 1 , , , , , over T f ( t ) are ellipses. The first four are given by: X : x c + ( a + c ) y b c − X : 9 x a + 9 y b − X : x ( a − c ) + y ( a + c ) b ( a − c ) − X : 4 x ( a − c ) + 4 ( a + c ) y a b − Remark . The vertices of the locus of X are f , f . Remark . At a = 2 c , i.e., a/b = 2 / √ ≃ . X is the verticalsegment [0 , ± b/ P ( t ) is at the top or bottom vertexof E , T f ( t ) is equilateral.Recall Line X X Nagel Line [13]. For the entire 40k+ centersin [12], the following 62 are fixed linear combinations of X , X (boldface indicatesthose in Proposition 15): , , , , , , 1125, 1698, 3241, 3244, 3616, 3617,3621, 3622, 3623, 3624, 3625, 3626, 3632, 3633, 3634, 3635, 3636, 3679, 3828, 4668, 4669,4677, 4678, 4691, 4701, 4745, 4746, 4816, 5550, 9780, 15808, 19862, 19872, 19875, 19876, Figure 10.
Left: elliptic loci of X k , k = 1 , , ,
10 (collinear on the Nagel line, dashed black)for a triangle family (dashed red) with two vertices on the foci. animation 1
Right: at a/b =2 / √ ≃ . a/b , the locus of X (blue) is a vertical segment. animation 2 .Note that X is the anticomplement of itself. Note also that X (resp. X ) isthe anticomplement of X (resp. X ). Numerically analyzing the loci of all 38k+on [12] which are not on the X X line we found none which produced an ellipseover T f ( t ). In turn this leads to the following conjecture: Conjecture 1.
The locus of X k over T f ( t ) = f f P ( t ) is an ellipse iff X k is afixed linear combination of X and X . Appendix D. Table of Symbols
Symbols used in the article appear on Table 3. symbol meaning E , a, b base ellipse and its semi-axes O, f , f center of foci of E V , V points fixed on E at t , t P ( t ) moving 3rd vertex of TT ( t ) E -inscribed triangle V V P ( t ) L e Euler line X X X ρ , ρ X ρ = X + ρ ( X − X ) c , d a − b , and a + b , resp. z shorthand for cos( t + t ) O ρ center of elliptic locus of X ρ L ρ , L k linear locus of O ρ over ρ (resp. V V parallels)∆ t envelope of L ρ for fixed V over V on E Γ ρ elliptic locus of O ρ for fixed V over V on E O ′ ρ , Γ ′ ρ center of Γ ρ and its elliptic locus over all V on E X , X incenter, barycenter X , X circumcenter, orthocenter X , X X , X Spieker center and X X midpoint Table 3.
Symbols used in the article.NVARIANTS OF CENTERS ELLIPSE-INSCRIBED TRIANGLES 17
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