On Vivianis Theorem and its Extensions

On Viviani’s Theorem and its Extensions Elias Abb oud Beit Berl College, Doar Beit Berl, 44905 Israel Email: eabb oud @b eitb erl.ac.il No v em ber 26, 2024 Abstract Viviani’s theorem states that the sum of distances from an y p oint in- side an equilater al triangle to its sides is constan t. W e consider extensions of the theorem and show that an y convex polygon can b e divided in to par- allel segmen ts such that the sum of the distances of th e p oints to the sides on eac h segment is constan t. A p olygon p ossesses the CVS prop erty if the sum of the distances fro m an y inner p oint to its sides is constant. An amazing result, concerning the conve rse of Viviani’s theorem is deduced; Three n on-collinear points whic h ha ve equal sum of distances to th e sides inside a conv ex p olygon, is sufficien t for p ossessing th e CVS prop erty . F or concav e p olygons the situation is quite different, while for p olyh ed ra analogous results are deduced. Key wor ds: distanc e sum function, CVS pr op erty, isosum se gment , isosum cr oss se ction. AMS subje ct classific ation : 51N20, 51F99, 90C05. 1 In tro duction Let P b e a p o ly gon or p olyhedro n, consisting of b oth b oundary and interior po int s. Define a distance sum function V : P → R , where for ea ch point P ∈ P the v alue V ( P ) is defined as the sum of the distances from the point P to the sides (faces) of P . W e say that P ha s the c onst ant Viviani su m prop erty , a bbreviated b y the ” CVS prop erty”, if and only if the function V is constant . Viviani (1622- 1703 ), who was a student a nd a ssistant of Galileo, discovered the theorem whic h states that equilateral triangles hav e the CVS prop erty . The theorem c a n be easily pr ov ed by an area argument; Joining a p oint P inside the triangle to its vertices divides it int o three parts, the sum of their are a s will b e equal to the area of the o riginal one. Therefore, V ( P ) will b e equal to the height of the triangle and the theorem follows. The imp ortance of Viviani’s theorem 1 may be derived from the fact that his teacher T orricelli (16 08-16 47) used it to lo cate the F ermat p oint of a triang le [2, pp. 443]. Samelson [6, pp. 225] gave a pro of of Vivia ini’s theo rem that us e s vectors and Chen & Liang [1, pp. 390-3 91] used this vector method for proving the conv erse of the theor em; If inside a tria ngle there is a circular reg ion in which V is consta nt then the triangle is equilatera l. Kaw asaki [3, pp. 213], by a pr o of without words, uses only rotations to establish Viviani’s theorem. There is an extensio n of the theorem to all regular po lygons, by the ar ea metho d: All regular p oly gons hav e the CVS pr op erty . There is also a n extension of the theorem to regular p olyhedro ns, by a volume argument: All regular p olyhedra hav e the CVS prop er t y . Ka wasaki, Y ag i and Y anag aw a [4, pp. 283] gav e a different pro of for the r egular tetrahedron. What happ e ns for ge neral p olyg ons and p o lyhedra? Surprisingly , there is a strict cor r elation betw een Viviani’s theorem, its co n- verse and ex tensio ns to linear progra mming. This correla tion is manifested by the following main result: Theorem 1.1 (a) A ny c onvex p olygon c an b e divide d int o p ar al lel se gment s such that V is c onstant on e ach se gment. (b) Any c onvex p olyhe dr on c an b e divid e d int o p ar al lel cr oss se ctions such that V is c onstant on e ach cr oss se ction. These segments or cross s ections, on which V is constant, will b e called iso- sum layers ( or definitely isosum se gments a nd isosum cr oss se ctions) . They are formed by the intersection o f P and a suitable family of parallel lines (planes). The v a lue o f the function V will incr ease when passing, in some direction, from one isosum layer in to another, unless P has the CVS proper ty . The correlation soo n will b e clea r. Each linear progra mming problem is comp osed of an obje ctive function and a fe asible r e gion (see for exa mple [5] o r [7]). Moreo ver, the ob jective function divides the feasible re g ion into isopr ofit lay ers, these layers a re par allel and consist of po int s on which the ob jectiv e function has constan t v alue. F urthermore, moving in some directio n will increase the v alue of the ob jective function unless it is constant in the feasible region. Because of this correlatio n we th us conclude the following amazing result, concerning the c o nv erse of Viviani’s theorem. Theorem 1.2 (a) If V takes e qual values at t hr e e non-c ol line ar p oints, inside a c onvex p olygon, then t he p olygon has the CVS pr op erty. (b) If V takes e qual values at four non-c oplanar p oints, inside a c onvex p oly- he dr on, then the p olyhe dr on has the CVS pr op erty. The theorem tells us that measuring the distances from the sides o f three non-collinear points, inside a conv ex p olygon, is sufficient for determining if the po lygon has the CVS proper ty . Likewise, measuring the distance s from the faces o f four non-coplana r points, inside a conv ex po lyhedron, is sufficient for determining if the p olyhedr on po s sesses the CVS prop erty . W e then end with the following b eautiful conclusions. 2 Corollary 1.3 (a) If ther e is an isometry of the plane which fixes the p olygon but n ot an isosum se gment, then the p olygon has the CVS pr op erty. (b) If a c onvex p olygon p ossesses a ro tational symmetry, ar ound a c ent r al p oint, then the p olygon has the CVS pr op erty. (c) If a c onvex p olygon p ossesses a r efle ction symmetry acr oss an axis l , then the p olygon has the CVS pr op erty or otherwise the isosum se gments ar e p erp endicular t o l . While for polyhedr a we hav e, Corollary 1.4 (a) If t her e is an isometry of the sp ac e which fix es t he p olyhe dr on but n ot an isosum cr oss se ction, then t he p olyhe dr on has the CVS pr op erty. (b) If a c onvex p olyhe dr on p ossesses two ro tational symmetries, ar ound dif - fer ent axes, then the p olyhe dr on has the CVS pr op erty. F rom Coro llary 1.3, o ne ca n deduce that all reg ula r p olygons hav e the CVS prop erty . Besides, an y parallelog ram has this prop erty , since it po ssesses a rotational symmetry around its centroid by a n angle o f 180 ◦ . Obviously , the existence o f tw o reflection sy mmetries , b y different axes, of a po lygon will imply a rotational symmetry and hence the po lygon must o wn the CVS prop erty . Moreov er, for triangles and quadrilater a ls, the existence o f a rotational sym- metry characterizes the p ossessing of CVS prop erty , s ince in these cases the po lygons w ould be only equilater a l tr iangles a nd parallelogr ams. Consequently , an n -gon, fo r n ≥ 5 , that does not p ossess the CVS pro p e r ty m ust hav e at most one symmetry which is the reflection symmetry . Analogously , b y Coro llary 1.4, all reg ula r po lyhedra a nd regular prisms ha ve the CVS prop erty . Likewise, any para llelepiped ha s the CVS pro pe r ty . Since it pos sesses three r otational symmetries b y an angle of 1 80 ◦ , around the a xes, each passing through the cen troids of a pair of parallel faces. On the other hand, the prop er t y of p oss e s sing a sy mmetry do es not c harac - terize a ll polygo ns (p olyhedra) satisfying the CVS prop erty . In section IV. w e will v alidate the existence of a po ly gon with only one r e flec tion s y mmetry , an asymmetric polygo n and a po lyhedron with a reflection symmetry only , which po ssess the CVS prop erty . W e will pro ceed as follo ws. In s e c tion I I., we first in tro duce a linear pr o- gramming problem for g eneral tria ngles. The main statement will b e; ”A tria ngle has the CVS prop e r ty if and only if it is equilateral, if and only if ther e are three non-collinear p o int s inside the triangle that hav e equal s um of dista nce s fro m the sides” . Then we dea l with gener al conv ex p olyg o ns and p olyhedro ns a nd the pr o of of theorems (1.1), (1 .2). Her e we shall rely on metho ds from ana lytic geometry beca use the use of co or dina tes. This allows us to determine the line (plane) which the is osum lay ers are para llel to. In section I I I. w e see what happ ens for co ncave p olygons and p olyhedr a, then in section I V. we co mpute some examples. 3 It is worthy men tioning that these results can b e stated and generalized for n − dimensional g eometry . 2 Con v ex P olygons and P olyhedra 2.1 The case of triangle: linear programming approac h Given a triang le △ AB C let a 1 , a 2 , a 3 be the lengths of the sides B C, AC , AB resp ectively . Let P b e a point inside the triang le a nd let h 1 , h 2 , h 3 be the distances (the leng ths of the altitudes) of the p o int P to the three sides resp ectively , see Figure 1. F or 1 ≤ i ≤ 3 , let x i = h i P 3 i =1 h i , where a s previously defined, P 3 i =1 h i = V ( P ). Clearly , for eac h 1 ≤ i ≤ 3 , w e have 0 ≤ x i ≤ 1 and P 3 i =1 x i = 1 . Denote x = ( x 1 , x 2 , x 3 ) and co nsider the linear function in three v ariables F ( x ) = P 3 i =1 a i x i . Now, this function is close ly rela ted to the function V . Accurately , F ( x ) = P 3 i =1 a i x i = P 3 i =1 a i h i P 3 i =1 h i = 2 S V ( P ) , where S is the ar ea o f the triangle. Consequently , F ( x ) = P 3 i =1 a i x i takes equal v alues in a subset of p oints of the feasible region if and only if the function V tak es equal v alues at the corresp o nding p oints inside the triangle. Thu s we may define the following linear prog ramming problem; The ob jective function is: F ( x ) = 3 X i =1 a i x i sub ject to the following constraints:  P 3 i =1 x i ≤ 1 x i ≥ 0 , 1 ≤ i ≤ 3 . Now, solving the pro blem means maximizing or minimizing the ob jective function in the feasible reg ion, and this optimal v alue must o ccur at some co r ner po int . F or that, o ne may use the simplex metho d, whic h is simple in this case beca use the simplex tableau cont ains only t w o r ows. But rather than using this algebraic metho d, and s ince we a r e not seeking for optimal v alues , and for better understanding the correla tio n with Vivia ni’s theorem, we use a geometr ic method. The feasible region will b e the right p yramid with v ertex (0 , 0 , 1) and basis vertices (0 , 0 , 0) , (1 , 0 , 0) , (0 , 1 , 0 ) and the iso profit pla nes are obtained by taking F ( x ) = P 3 i =1 a i x i = c, w her e c is co nstant, see Fig ure 2. These is oprofit planes will meet the face P 3 i =1 x i = 1 across a line which in its turn will corr esp ond to an iso sum segment inside the triangle △ AB C. The v alue of the function V will b e co nstant for all p oints on the isosum seg- men t. Mor eov er, moving in a p erp endicular direction to the isosum segments will increase the v alue of V . Hence, V do es no t tak e the same v alue at an y three non-collinear p oints inside the triangle. 4 This is true in general, except for one case where the isopr ofit planes , P 3 i =1 a i x i = c, are parallel to the face P 3 i =1 x i = 1 . This exceptional cas e o ccurs if and only if the vectors ( a 1 , a 2 , a 3 ) and (1 , 1 , 1) are linea r ly dep endent i.e., ( a 1 , a 2 , a 3 ) = λ (1 , 1 , 1 ) , and this happ ens if and only if the tria ngle is equilateral. Thus proving the following theorem. Theorem 2.1 (a) Any triangle c an b e divide d into p ar al lel se gmen t s such that V is c onstant on e ach se gment. (b) The fol lowing c onditions ar e e quivalent • The triangle △ AB C has the CVS pr op erty. • Ther e ar e thr e e non-c ol line ar p oints, inside the triangle, at which V takes the same value. • △ AB C is e quilater al. 2.2 Extension for con v ex polygons: analytic geometry ap- proac h Inspired by the geometric metho d of the linear progr a mming problem defined in the previous s ubse c tion, we attack the general case by using analytic geometry tech niques. Firstly , we aim to prove Theorems 1.1(a) and 1 .2(b). Given a p olygon with n sides, we embed it in a Ca rtesian plane. Suppo se that its sides lie on lines with equations; α i X + β i Y + γ i = 0 . (1) Since the con vex p olygon lies on the same side o f line (1) then the express ion α i x + β i y + γ i has unc hanging sign, for all P = ( x, y ) inside the po ly gon. Thu s the distance, h i , of the p oint P from ea ch side o f the p olygo n is given b y the equation; h i = ( − 1 ) ε i α i x + β i y + γ i q α 2 i + β 2 i , where ε i ∈ { 0 , 1 } . Therefore, the function V is given by a linear express ion V ( x, y )= n X i =1 ( − 1) ε i α i x + β i y + γ i q α 2 i + β 2 i . (2) 5 Letting this sum equals some constant c, we get the equation of the line which the is osum segments are parallel to; n X i =1 ( − 1) ε i α i x + β i y + γ i q α 2 i + β 2 i = c. (3) If the p oint P = ( x, y ) is restr ic ted to b e inside the polygo n then for different v alues o f c, we get para llel segments. On ea ch such s e g ment , the function V takes the constant v alue c . This pr ov es Theorem 1.1 (a). The equation g iven b y (2) is indep endent of ( x, y ) if and only if the v ariable part of the function V v anishes. But then V is constan t and so the p olygon has the CVS prop erty . If there a re three non-co llinear po int s inside the p olygon at which V tak es the sa me v a lue then there exist t wo differen t isosum segments on which the function V takes the same v alue. This happ ens if and only if V is constant and so the p o ly g on once aga in has the CVS pro p er t y . Hence Theorem 1.2(a) is prov ed. W e turn now to proving Coro llary 1.3. (a) If there is an isometry of the plane which fixes the polyg on but not some isosum la yer, then this will assur e the existence of three non-collinear po int s inside the p o ly gon at whic h V takes e q ual v alues. That’s becaus e an isometry preserves distances and the sets of boundary and inner p oints. Th us Theorem 1.2(a) yields that the p o lygon ha s the CVS pro p er t y . (b) F ollows from part (a). (c) If the isosum segments are not fixed by the r eflection, then accor ding to part (a) the polygo n has the CVS prop erty . Otherwise, if fixed, then the isos um segments must be p er pendicula r to the reflec tio n a x is. 2.3 Extension for con v ex p olyhedra The pro o f for a Polyhedron is s imilar up to minor modifica tions. The faces lie on planes with equa tions; α i x + β i y + γ i z + δ i = 0 , and the linear function V b ecomes; V ( x, y , z ) = n X i =1 ( − 1) ε i α i x + β i y + γ i z + δ i q α 2 i + β 2 i + γ 2 i , where ε i ∈ { 0 , 1 } . The same argument yields the result for the p olyhedr o n (part (b) of Theo- rems 1.1 and 1 .2). The pro of of Corollary 1.4 is similar to cor ollary 1.3; (a) F ollows fr o m pa rt (b) of Theo rem 1.2. (b) T wo ro tational symmetries o f a p olyhedro n aro und different axes, gua r - antee the ex istence of four non-c o planar points inside the polyhedron at which V takes equal v alues. By Theorem 1 .2 (b), the result follows. 6 3 Conca v e P olygons and P olyhedra The situation for co ncav e p olygons a nd p olyhedra is quite diff erent. Theo rems 1.1 and 1.2 ar e no longer v alid. Moreover, concave p olygons and po lyhedra don’t hav e the CVS prop erty . Surprisingly , with a little mo re elab o r ate effor t, one might r ather find a ge ner ality of Theorem 1.1. Concerning polygons, the crucia l po int which makes the difference is that the points inside a concav e po ly gon are no lo nger lie o n the same side of each b o undary line. This was a k ey for defining the distance s um function V . W e turn to giv ing an example to illustrate the theme. Let AB C D b e the con- cav e k ite with vertices (0 , 8) , ( − 6 , 0) , (0 , 2 . 5) , (6 , 0) r esp ectively . Let l 1 , l 2 , l 3 , l 4 be the lines containing the sides AB , B C, C D, D A and E , F b e the int ersection po int s of l 1 , l 3 and l 2 , l 4 resp ectively , see Figur e 3. Then the co ncave kite AB C D is divided into three distinct conv ex p o lygonal regions, namely; AE C F , E B C and F C D . All p oints inside an y r egion lie on the same side of each l i , 1 ≤ i ≤ 4 . T o explain that, note that any line l divides the plane in to t w o half-planes ” O l ” which contains the origin and its complemen t ” O c l ” . The following table shows the lo cation of the p oints inside a n y region relative to the lines l i , 1 ≤ i ≤ 4 . lo cation relative to region AE C F E B C F C D l 1 l 2 l 3 l 4 O l 1 O c l 2 O c l 3 O l 4 O l 1 O c l 2 O l 3 O l 4 O l 1 O l 2 O c l 3 O l 4 Note that an y t wo neig hboring regio ns, having a common edg e, differ only in one en try in this table. F or instance, the po int s of AE C F and E B C lie on the same side of l 1 , l 2 , l 4 and on o ppo site sides of l 3 . The implem entation is that the distance sum function V will be a split function compo sed of three components; V AB C D ( P ) =    V AE C F ( P ) , P inside AE C F V E B C ( P ) , P inside E B C V F C D ( P ) , P inside F C D , where each co mpo nen t is given by a linear expression as in equation (2), definitely; 7 V AE C F ( P ) = − − 8 x + 6 y − 4 8 10 + − 5 x + 12 y − 30 13 + 5 x + 12 y − 30 13 − 8 x + 6 y − 48 10 V E B C ( P ) = − − 8 x + 6 y − 4 8 10 + − 5 x + 12 y − 30 13 − 5 x + 12 y − 30 13 − 8 x + 6 y − 48 10 V DC F ( P ) = − − 8 x + 6 y − 4 8 10 − − 5 x + 12 y − 30 13 + 5 x + 12 y − 30 13 − 8 x + 6 y − 48 10 This s ays that each region ca n b e divided int o pa rallel isosum se gments but in a different direction. In our example, the isosum segments of AE C F a re parallel to a line of the form y = c, the isosum segment s of E B C are par allel to the line 100 x + 156 y = 0 and the isosum segments of D C F are par allel to the line 1 0 0 x − 156 y = 0 . Moreov er, t wo isosum segments from neighbor ing reg ions which meet a t a p oint o n the common edge, define three non-collinear po int s with the same distance sum fro m the sides of AB C D , s ee Figure 3. These ideas can b e genera lized to a ny concav e polyg on or p olyhedr on. Th us we hav e the following theorem; Theorem 3.1 (a) Any c onc ave p olygon c an b e divi de d int o c onvex p olygonal re - gions such that e ach r e gion c an b e divide d into p ar al lel isosum se gments. Mor e- over, isosum se gments of neighb oring r e gions have differ ent dir e ctions. (b) Ther e ar e thr e e non-c ol line ar p oints inside a c onc ave p olygon which have e qual distanc e sum fr om the sides. (c) Any c onc ave p olyhe dr on c an b e divide d into c onvex p olyhe dr al r e gions such that e ach r e gion c an b e divide d into p ar al lel isosum cr oss se ctions. Mor e over, isosum cr oss se ctions of neighb oring r e gions have differ ent dir e ctions. (d) Ther e ar e four non-c oplanar p oints inside a c onc ave p olyhe dr on which have e qual distanc e sum fr om the fac es. (e) Conc ave p olygons and p olyhe dr a do not p ossess the CVS pr op ert y. Pr o of. We shal l outline the pr o of for p olygons. (a) L et P a c onc ave p olygon. Extend the s ides and c onstruct al l p ossible interse ct ion p oints in P of the b oundary lines, se e Figur e 4. In this way one gets a p artition of P into m c onvex p olygonal r e gions P 1 , P 2 , ..., P m . Note that e ach side of any P j lie on a b oundary line of P . The p oints inside any two neighb oring r e gions P i , P j with a c ommon e dge LM , lie on t he same side of e ach b oundary line of P exc ept one, namely, the b oundary line l LM c ont aining LM , se e Figur e 4 . We claim t hat t he c onjugation of these two neighb oring r e gions P i ∪ P j , wil l r emain a c onvex p olygonal r e gion. This is true sinc e the t wo sides of P i and P j which me et at L (or M ) lie on the same b oundary line of P , by the c onstru ction (in Figur e 4 the p oints K , L, D ar e c ol line ar, likewise I , M , D 8 ar e c ol line ar). Thus al l p oints of P i ∪ P j lie on the same side of the b oundary lines of P exc ept l LM . Now, the distanc e sum function V o f P is define d as a split function; V P ( P ) =            V P 1 ( P ) , P inside P 1 V P 2 ( P ) , P inside P 2 . . V P m ( P ) , P inside P m . The p oints on a c ommon e dge of neighb oring r e gions c an b e attache d to any one of b oth. Each c onvex p olygo nal r e gion has p ar al lel isosum se gments ac c or ding to the line ar expr ession which defines V P i , 1 ≤ i ≤ m. Isosum se gments of neighb oring r e gions P i , P j have differ ent dir e ctions sinc e the line ar expr essions of V P i , V P j differ only in one sign as explai ne d ab ove. (b) T ake any p oint P on a c ommon e dge of two neighb oring r e gions P i , P j and let s 1 , s 2 b e two isosum se gments of P i , P j r esp e ct ively, issuing fr om P . L et Q 1 , Q 2 two p oints on s 1 , s 2 r esp e ct ively. Sinc e s 1 , s 2 have differ ent dir e ctions then P , Q 1 , Q 2 ar e non-c ol line ar and have the same distanc e sum fr om the sid es. (e) The distanc e sum functions V P i , V P j of t wo neighb oring r e gions P i , P j have t he same line ar expr essions with the same signs exc ept one; V P i = m P i =1 ( a k x + b k y + c k ) + ( a 0 x + b 0 y + c 0 ) and V P j = m P i =1 ( a k x + b k y + c k ) − ( a 0 x + b 0 y + c 0 ) . Thus V P c annot b e c onstant on P . 4 Examples In the followin g w e co mpute the equations o f the is osum lay ers for pa rticular po lygons and p olyhedra. It is an easy matter to chec k the re s ults, b y co nstruct- ing the figures using a computer pro gram such as the geometer sketc hpad or wingeom. Example 4.1 (Kite) Construct a kite in the Cartesian plane with vertices, A = (0 , β ) , B = ( − α, 0) , C = (0 , γ ) , D = ( α, 0) , where, α, β > 0 a nd γ < 0 . The equations of the lines containing the sides AB , B C, C D , D A, respec- tiv ely are; − αy + β x + αβ = 0 − αy + γ x + α γ = 0 αy + γ x − αγ = 0 αy + β x − αβ = 0 . 9 Thu s the function V is g iven b y; V AB C D = − αy + β x + αβ p α 2 + β 2 − − αy + γ x + α γ p α 2 + γ 2 + αy + γ x − αγ p α 2 + γ 2 − αy + β x − αβ p α 2 + β 2 . Equiv a le ntly , V AB C D = − 2 αy p α 2 + β 2 + 2 αβ p α 2 + β 2 + 2 αy p α 2 + γ 2 − 2 αγ p α 2 + γ 2 . (4) As a result we see that the isosum s egments ha ve simple equations, na mely , y = c. Obviously , equation (4), shows that the V AB C D is indep endent of v ariables if and only if β = γ . This means that the p olygon is in fact a para llelogram and hence, it owns the CVS prope r ty . On the other side, if β 6 = γ , then the kite do es not own the CVS pr op erty , but it can be divided into isosum seg ment s which a re para llel to the diagona l B D , whic h is consistent with Cor ollary 1.3 (c). Example 4.2 (Isosc eles triangle) In the previo us example if one considers P as a p oint inside the isosceles triangle AB D then, V AB D = − αy + β x + αβ p α 2 + β 2 − αy + β x − αβ p α 2 + β 2 + y . Equiv a le ntly , V AB D = ( p α 2 + β 2 − 2 α ) y p α 2 + β 2 + 2 αβ p α 2 + β 2 . The function V AB D is indep e ndent of v ariables if and only if β = √ 3 α. In which case the triangle is in fa ct equilateral and hence it has the CVS prop erty . If β 6 = √ 3 α, then it can b e divided in to isos um segmen ts parallel to the base, which is consistent with Corollar y 1.3 (c) Example 4.3 (Qu adrilater al) Let b e given the quadrilateral with vertices (0 , 0 ) , (3 , 0) , (1 , 2) , (0 , 1) . Direct computations will show that the isosum segments are parallel to the line y = (1 + √ 2) x, see Figure 5. This quadr ila teral do es not have the CVS prop er t y . Example 4.4 (Pentagon with a r efle ct ion symmetry) 10 The p entagon AB C D E with vertices A = (0 , 3 + √ 3) , B = ( − 1 , 3) , C = ( − 1 , 0) , D = (1 , 0) , E = (1 , 3) and sides 2 , 3 , 2 , 3 , 2 resp ectively , owns the CVS prop erty a lthough it has one reflection symmetry a nd no rotational symmetries. It is a stra ightf orward matter, to show the p ossessing of the CVS prop erty , s ince it is co mpo sed of a rectangle a nd an equilateral triangle. In g eneral, supp ose that AB C DE is a p entagon where AB E is a n equilatera l triangle with side-length a and height h, a nd B C D E is a rectangle with side- lengths a, b. Then the distance sum of an inner p o int from its sides is a + b + h. This is obvious if the p oint is inside the triangle. If the p oint is inside the rectangle then observe that it is lo cated on the base o f another equilateral triangle o btained by extending the sides AB , AE and the result easily follows. Example 4.5 (Asymmetric p ent agon) Construct any pentagon AB C D E with a ng les 70 ◦ , 110 ◦ , 130 ◦ , 60 ◦ and 170 ◦ . Here a g ain it is a straig ht forward matter, to show that AB C D E poss ess the CVS prop erty , since it is comp osed of a para llelogram and an equilateral triang le. This p entagon has no sy mmetries. Observe that in this case, the distance sum of a n inner p oint from the sides is a + b + h, w her e a, b are the distances betw een the opp osite sides of the parallelog ram and h is the height of the equilater al triang le. Example 4.6 (Equiangular p olygon) An y equiangular p olyg on P with n s ides has the CVS pro per ty . This can b e demonstrated by the follo wing argument; Lo cate inside the equia ngular p olygon a re g ular n -g o n, P r . Ro tate P r around its centroid until one of its sides gets parallel to one side of P . But then all corresp onding sides o f b oth p olygo ns get parallel. Let V P , V P r be the distance sum functions defined on P and P r resp ectively . Then for any point P inside P r we hav e, V P ( P ) = V P r ( P ) + c, where c r e pr esents the sum o f distances b etw een the par allel sides of P and P r . Since V P r is constant inside P r and also c is cons ta nt then V P is consta n t inside P r . By Theorem 1.2(a), V P is constant on P . Example 4.7 (Polyhe dr on with a r efle ction symmetry) Construct the rig ht prism for which the p entagon AB C D E from the prev io us example co ngruent to its bas e s . This prism has only one s y mmetry , namely , the reflection symmetry by the plane parallel to its t wo bases and passe s in the middle. This p oly hedr on has the CVS pro p e r ty . Since v ertically the sum of the t wo altitudes to the bases from any inner p o int is constant and equals to the height of the prism and, horizontally , the sum of the a ltitudes to the faces is also co nstant b y the previous example. 11 Example 4.8 (Pyr amid with a r otational symmetry ar ound one axis) Construct in the space , the rectang ular pyramid with vertices A = (0 , 0 , α ), B = ( β , 0 , 0 ) , C = (0 , γ , 0) , D = ( − β , 0 , 0 ) and E = (0 , − γ , 0 ) , where α, β , γ > 0 . The face s AB C, AC D , AD E a nd AE B lie o n the following planes resp ectively; αγ x + αβ y + β γ z = α β γ − αγ x + αβ y + β γ z = αβ γ − αγ x − αβ y + β γ z = αβ γ αγ x − αβ y + β γ z = α β γ . Letting ∆ = p α 2 γ 2 + α 2 β 2 + β 2 γ 2 then the altitudes from the p oint P = ( x, y , z ) to the faces ar e given by; h B C DE = z h AB C = − 1 ∆ ( αγ x + αβ y + β γ z − α β γ ) h AC D = − 1 ∆ ( − αγ x + αβ y + β γ z − αβ γ ) h ADE = − 1 ∆ ( − αγ x − αβ y + β γ z − αβ γ ) h AE B = − 1 ∆ ( αγ x − αβ y + β γ z − α β γ ) Thu s V AB C DE = (1 − 4 β γ ∆ ) z + 4 αβ γ ∆ . Evidently , the isos um lay ers are parallel to the bas e of the pyramid a nd it has the CVS prop erty if and only if α = √ 15 β γ p β 2 + γ 2 . Figure 6 shows a sp e cial case where, β = γ = 1 and α = q 15 2 . 5 Concluding Remarks • Co rollar y 1.3 par t (c) c a n b e extended for po lyhedra but the prop erty of being p erp endicular to the reflection plane would not uniquely determine the isosum cross sections. • It is p ossible to state an algebraic neces s ary and sufficient condition for the CVS prop erty , using the expres sion of V given in (2 ). But a geometric one is more fav orable. Thu s the substantial question is how ca n one c haracterizes , geo metrically , all those p o lygons and p olyhedra that satisfy the CVS proper t y? 12 Ac kno wledgemen t 5.1 The author is indebte d for al l c ol le agues who to ok p art in r eviewing the manuscript. Sp e cial thanks ar e due to the anonymous r efer e es for their valuable c omments that impr ove d the exp osition of the p ap er. References [1] Z hib o Chen a nd Tian Liang, The Conv erse of Viviani’s Theorem, The Col- le ge Mathematics J ournal, V ol. 37, No. 5 (2006) , pp. 390-39 1. [2] Shay Gueron and Ran T essler, The F ermat-Steiner P roblem, Amer. Math. Monthly 109 (2002), pp. 443-451. [3] K en-ichiroh Kaw asaki, Pro o f Without W ords: Viviani’s Theorem , Mathe- matics Magazine,V ol. 78, No. 3 (2005), pp. 213. [4] K en-ichiroh K aw asaki, Y. Y agi and K. Y anagawa, On Viviani’s theorem in three dimensions, The Mathematic al Gazette, V ol. 89, No.51 5 (2005), pp. 283. [5] Ma rgare t L. Lia l, Raymond N. Greenw ell, Na than P . Ritchey , Finite Math- ematics, Eight Edition, Pearso n Education, Inc., 2005. [6] H. Samelson, Pr o of Without W ords: Viviani’s Theorem with V ectors , Math- ematics Magazine, V ol. 76, No. 3 (2003), pp. 225. [7] Ro ber t J. V anderb ei, L inea r Prog ramming; F oundations and Extensions , Third E dition, Springer, 2 008. Figure 1: A p oint P inside the tria ng le with dista nces h 1 , h 2 , h 3 . 13 Figure 2: The feasible region with a n isoprofit plane: F ( x ) = 0 . Figure 3: The isosum seg men ts have different directions and X , Y , Z are non- collinear points with equal distance sum fro m the sides of AB C D . 14 Figure 4: Partitioning the concav e p o lygon AB C D E F GH into conv ex p oly g o- nal r egions. Figure 5: The isosum segments are para llel to the line y = (1 + √ 2) x. 15 Figure 6: A squar e pyramid which has the CVS prope r ty . 16