Wednesday, September 18, 2013

Recursive functions with cyclic sums


RECURSIVE FUNCTIONS WITH CYCLIC SUMS

The recursive equalities shown here make easier working with cyclic sums applied to triangles, specially when a conversion of (a,b,c) to (R,r,s) is required, i.e, when some triangle data must be expressed as function of  R (circumradius), r (inradius) and s (semi-perimeter).
Let’s name

σ(n) = [ a­­ ]
ρ(n) = [ (b c)­­ ]
λ(k, m, n)= [ ak (bm cn + bn cm)­­­ ]   m>=n

where k, n, m are integer numbers and all sums are cyclic, and
Π(n) = (a b c)n = [Π(1)]n

General expression for σ(n):
It is obvious that σ( 0 ) = 3 and σ( 1 ) = 2 s and it is well known that σ(2)=a2 + b2 + c2  = s2 – r2 – 4 R r and   Π(1)=4*R*r*s.
For other side, if n<0 then
σ(n) = [ 1/a­­|n|­ ] = (a b c)n [ (b c)­­|n|­ ] = Π(n) ρ(-n)       [n<0]

 Suppose now that n>=3 and let’s calculate σ(1) σ(n-1), i.e.
σ(1) σ(n-1) = [ a­­ ] [ a­­n­-1 ]  =
σ(n) + a ( bn-1 + cn-1 ) + b ( cn-1 + an-1 ) + c ( an-1 + bn-1 ) =
σ(n) + b c ( bn-2 + cn-2 ) + c a ( cn-2 + an-2 ) + a b ( an-2 + bn-2 ) =
σ(n) + b c (σ(n-2) - an-2 ) + c a (σ(n-2) - bn-2) + a b (σ(n-2) - cn-2) =
σ(n) + σ(n-2) [ b c­­ ] – a b c σ(n-3) =
σ(n) + ρ(1) σ(n-2) – Π(1) σ(n-3)
from which it is possible to deduce σ(n) as:
σ(n) = [ a­­ ] =
Π(n) ρ(-n)                                                                  if n<0
3                                                                                if n=0
2 s                                                                             if n=1
s2 – r2 – 4 R r                                                           if n=2
σ(1) σ(n-1) - ρ(1) σ(n-2) + Π(1) σ(n-3)                    if n>=3

In a similar way it is easy to obtain expressions for ρ(n) ( for n>0 just get it from (an + bn +cn)2  ) and (with a little more effort) for  λ(k, m, n). These are:

ρ(n) = [ (b c)­­ ] =
Π(n) σ(-n)                                         if n<0
3                                                       if n=0
( σ(n)² - σ(2 n) )/2                            if n>0

and:

λ(k, m, n)= [ ak (bm cn + bn cm)­­­ ]  = Π(n) ( σ(m-n) σ(k-n) - σ(k+m-2 n) )    
for m>=n 

Note that σ(n) and ρ(n) call each one the other. It can be checked that this fact does not lead to any infinite loop.
Also note that these functions can be extended to other cyclic sums: you only need their cyclic product and the cyclic sums of the powers 0, 1 and 2 (…and power 0 is always equal to 3). For example, for working with σ(n) = ∑ [cos(A) ]n, ρ(n) = ∑ [cos(B) cos(C) ]n and λ(k,m,n) = ∑ [cosk(A) (…) ] you only need:
σ(1) = (R+r)/R,  σ(2) = (-s2+6R2+4Rr+r2)/(2*R2) and Π(1)= (s2-4R2-4Rr-r2)/(4*R2)
The implementation of these functions in Maple and a small procedure (which determines what function and parameters to use) gave me, in half-one-minute, the 48 combos visible by pressing the next button:
    


NOTE ADDED Sept. 18, 2013-21:35: Barry Wolk kindly informed me that Maple can do this same work just using "Simplify with side relations". I appreciate this tip very much, which I ignored. Thank you, Barry.

Wednesday, September 11, 2013

Conic through 40 ETC centers - Locus

The circumconic U with trilinear equation:
U :  [ (b^2 – c^2)* v * w ] = 0
having center X(244) and perspector X(661) passes through A, B, C and these 40 ETC centers: X(1), X(10), X(19), X(37), X(65), X(75), X(82), X(91), X(158), X(225), X(267), X(596), X(759), X(775), X(876), X(897), X(921), X(969), X(994), X(1247), X(1581), X(1910), X(2153), X(2154), X(2166), X(2168), X(2186), X(2190), X(2214), X(2216), X(2217), X(2218), X(2219), X(2363), X(2588), X(2589), X(2652), X(2962), X(3668), X(4674).
U is the locus of P (not on the circumcircle of ABC) such that the antipedal triangle of P w/r to a ABC and the incentral triangle of ABC are orthologic.
X(759) is the fourth intersection of U and the circumcircle.
If O1(P) is the orthologic center (Incentral ; P-Antipedal ) for P then ( P, O1(P) )=( X(I), X(J) ) for these (I,J): (1,1), (10, 4065), (19,4319), (37, 2667), (65,2292), (75,192), (596,3159), (759,5497)
Some non-ETC O1(P) are:
P O1(P) Details of O1(P)
X(82) (a^2+b^2+c^2-a*(b+c)+b*c)*(a^2+b^2-b*c+c^2) On lines: (1,2896), (37,82), (192,3938), (744,2667), 3057,3100), (3159,3685), (4085,5262), (4514,4972)
X(876) (a^2-b*c)*(b^2+c^2-a*(b+c))*(b-c) Reflection of X(I) on X(J) for these (I,J): (2254,665), (3766,3716)
On lines: (1,514), (37,513), (190,5378), (192,522), (512,2292), (523,2667), (649,4414), (659,4435), (665,1642), (764,4983), (891,3251), (1281,2785), (1742,3667), (3057,4083), (3716,3766), (4065,4151), (4124,4448)
X(897) (a^2+b^2+c^2-3*b*c)*(a^2+b^2+c^2-3*a*(b+c)+3*b*c) On lines: (1,2796), (37,100), (2836,3057), (4442,4956)
X(969) (a^2+b^2+c^2+4*b*c)*(a^2+b^2+c^2+2*a*(b+c)) On lines: (1,193), (37,63), (192,612), (975,3159), (1773,3743), (2667,3870), (3920,4319), (4461,5297), (4657,5241)
X(994) (a-2*(b+c))*(b^2+c^2+a*(b+c)-b*c) On lines: (1,89), (2,4674), (8,3159), (37,517), (42,3899), (45,4752), (145,4065), (190,996), (192,519), (386,3878), (514,1000), (750,3245), (758,2667), (982,3898), (984,2802), (986,3884), (991,2800), (995,3877), (2099,4653), (2177,4867), (3670,3890), (3679,4125), (3727,3730), (3938,5497), (4256,5289)
X(2214) (a^2+(b+c)^2)*(a^2+b^2+c^2+a*(b+c)) On lines: (1,69), (31,37), (192,3920), (344,1961), (534,4319), (612,2345), (1486,1962), (2667,3938), (3057,4336), (3159,3923), (3966,4657)

The locus of O1 is the conic V1 with trilinear equation:
V1: ∑ [ a*(b-c)*(u^2 + v*w) ] = 0
The center of V1 is:
   (b+c)*(a*(b+c)*(a^2-b*c)+(b^2-4*b*c+c^2)*a^2+2*b^2*c^2) : :
and lies on lines: (1,4427), (42,3952), (244,1962), (659,3722), (740,899), (1193,4065), (2292,2611), (2667,4117), (2802,3743)
V1 passes through the vertex of the incentral triangle and these 10 ETC centers: X(1), X(37), X(192), X(2292), X(2667), X(3057), X(3159), X(4065), X(4319), X(5497) .

Some non-ETC P (given O1(P)):
O1(P) P Details of P
X(37) 1/(a^2-a*(b+c)-b*c) Isogonal conjugate of X(1621) 
Midpoint of (3555,3696) 
On lines: (10,141), (37,38), (65,1418), (75,3873), (81,82), (225,1876), (244,872), (596,740), (674,3664), (692,3449), (876,4132), (1002,4000), (1037,5228), (1468,2218), 1486,3423), (3286,3941), (3446,5091), (3555,3696), (3668,5173), 3681,4751), (3742,4698), (3779,4675), (4032,5083), (4430,4699)
X(3057) 1/(a^3-(b^2-b*c+c^2)*a-b*c*(b+c)) Isogonal conjugate of X(2975)
On lines: (1,859), (5,10), (28,2190), (31,2217), (37,1953), (48,2214), (60,1610), (65,1193), (72,4692), (75,3869), (143,952), (197,1036), (214,5482), (225,1829), (595,759), (596,758), (957,3086), (961,3450), (1460,3435), (2218,3915), (2390,4292), (2933,5264), (3216,4674), (3668,3827)

The orthologic centers O2(P) (P-Antipedal ; Incentral) have complicated expressions. The only ETC-defined correspondence is O2( X(1) ) = X(3). The locus of O2(P) is also a conic with center of few interest.

Sunday, September 8, 2013

Interesting locus: Conic through 34 ETC centers


What is the locus of P such that the anticevian triangle of P w/r to ABC is perspective to the intangents triangle of ABC?
The answer is:
1) The antiorthic axis of ABC, line (X(44)X(513)), with trilinear equation u+v+w = 0, or
2) The circumhyperbola U with trilinear equation
U: ∑ [ a*(b-c)*(b+c-a)*v*w ] = 0
The perspector of U is X(663).
The center of U is:
a*(b-c)^2*(b+c-a)*(a^3-2*(b+c)*a^2+(b^2+b*c+c^2)*a-b*c*(b+c)) : :
and lies on lines (X(I)X(J)) for these (I,J): (41,4557), (649,4014), (657,3271), (1200,2246), (3124,5075)      
U pasess through A, B, C and these 34 ETC centers: X(6), X(9), X(19), X(55), X(57), X(284), X(333), X(673), X(893), X(909), X(1024), X(1174), X(1436), X(1751), X(1945), X(2160), X(2161), X(2164), (2195), X(2258), X(2259), X(2291), X(2299), X(2316), X(2319), X(2337), X(2339), X(2343), X(2364), X(2432), X(2590), X(2591), X(3451), X(3512) 

Mappings:
1) If P is on the antiorthic axis of ABC then the perspector Z(P) lies on line (X(650)X(663)) with trilinear equation: u/(s-a)+v/(s-b)+w/(s-c)=0. This line passes through this 15 ETC-centers: X(650), X(663), X(861), X(2340), X(3689), X(3900), X(4041), X(4105), X(4162), (4433), X(4435), X(4477), X(4814), X(4895), X(4959)
There are 92 known ETC centers on the antiorthic axis (up to X(5546)) and for only two of them the correspondant perspectors are defined in ETC. These [ P, Z(P) ] are [X(650), X(4162)] and [X(652), X(663)].
You can see/download the mapping from this part of the locus in table 1 and table 1.2 at end of this page, [Note: most of pairs (P, Z(P)) are not easy to handle and only the most simple ones has been calculated.]

2) If P lies on conic U, then the perspectors Z(P) lie on the hyperbola V with trilinear equation:
V: ∑ [ a*(b-c)*(b+c-a)^2*v*w ] = 0
The perspector of V is X(657).
The center of V has trilinears:
a*(b-c)^2*(-a+b+c)^2*(a^2*(a^2-b^2+b*c-c^2)-a*(b+c)*(-b^2-c^2+a^2)-b*c*(b-c)^2) : :
or, equivalently:
cos(A/2)^2*(cos(2*A)+r/R+3)*sin(B/2-C/2)^2 : :
and lies on lines (X(I)X(J)) for these (I,J): (1,4566), (11,3138), (663,3022), (997,4319), (1015,3269), (1064,2293), (1984,2310)
The hyperbola V passes through A, B, C and these 14 ETC centers: X(1), X(33), X(55), X(64), X(103), X(200), X(220),X(963), X(1043), X(2192),X(2328), X(2332), X(2342), X(4845).
For P on cubic U, (P, Z(P)) = (X(I),X(J)) for these [I,J]:   [6, 55], [9, 200], [19, 33], [55, 220], [57, 1], [284, 2328], [333, 1043], [909, 2342], [1436, 2192], [2291, 4845], [2299, 2332] .
Please see/download tables 2.1 and 2.2  for complete lists of points and perspectors.

  Direct mapping for P on antiorthic axis   Download
  Inverse mapping for Z(P) on line (X(650)X(663))   Download
  Direct mapping for P on conic U   Download
  Inverse mapping for Z(P) on conic V   Download

Note: Show tables seems to work well only with Google Chrome.

Saturday, September 7, 2013

Nice trilinear coordinates

The sum and difference of two simple terms were applied to the ETC search triangle (6,9,13). The resulting numerical coordinates were sought in the ETC tables and then they were checked algebraically.

SA, SB, SC, SW are the standard variables for Conway notation.
S=2*area(ABC), R=circumradius, r=inradius and s=semiperimeter of triangle.

X(7) (b*c-SA)/a X(610) SB*SC-a^2*SA X(2223) a^2*(a*s-SW)
X(8) (b*c+SA)/a X(612) b*c+SW X(2285) SB*SC+a^2*b*c
X(9) a-s X(614) b*c-SW X(2300) a^2*(a*s+SA)
X(19) a^2*SA-S^2 X(615) a*R-b*c X(2308) a*(a*r+S)
X(37) a*r-S X(661) SB-SC X(2548) a^3*R+b*c*S
X(38) SA+SW X(672) a*(a*s-SW) X(2964) a^2-R^2
X(41) a^2*(a-s) X(748) a^2*r*s-b*c*S X(2965) a*(a^2-R^2)
X(42) a*(a*r-S) X(750) a^2*r*s+b*c*S X(2999) b*c-s^2
X(47) a^2*r*s-S*R^2 X(798) SB^2-SC^2 X(3053) a*(a^2-SA)
X(48) SB*SC-S^2 X(894) (b*c+a^2)/a X(3068) (S+a^2)/a
X(55) a*(a-s) X(940) a*s+b*c X(3069) (S-a^2)/a
X(57) SA-b*c X(968) a^2*r-S*s X(3083) b*c+S
X(58) a^2*s+S*R X(1100) a*r+S X(3084) b*c-S
X(63) SA X(1124) a*(S+b*c) X(3299) a+R
X(75) SA^2+S^2 X(1193) a*(a*s+SA) X(3301) a-R
X(77) SA*(SA-b*c) X(1203) a*s-R*r X(3553) a*R+r*s
X(78) SA*(SA+b*c) X(1267) (b*c+S)/a X(3553) a^2*b*c+S^2
X(171) a^2+b*c X(1335) a*(S-b*c) X(3554) a*R-r*s
X(191) SA+R*r X(1386) a*s+SW X(3554) a^2*b*c-S^2
X(193) (SA-a^2)/a X(1449) a+s X(3624) b*c+R*r
X(213) a^2*(a*r-S) X(1468) a*(a*s+b*c) X(3666) SA+a*s
X(238) a^2-b*c X(1572) a^3*s+S^2 X(3751) a*S-SW*r
X(239) (b*c-a^2)/a X(1580) a^4-b^2*c^2 X(3915) a*(a*s-b*c)
X(326) b^2*c^2-S^2 X(1582) a^4+b^2*c^2 X(4258) a*(a^2-s^2)
X(405) b*c*S+SA*r*a X(1698) b*c-R*r X(4383) a*s-b*c
X(474) b*c*S-SA*r*a X(1707) a^2-SA X(4512) a^2-s^2
X(491) (SA-S)/a X(1743) a*S-s^2*r X(4641) SA-a*s
X(492) (SA+S)/a X(1953) SB*SC+S^2 X(5019) a^2*(a*s+b*c)
X(518) a*s-SW X(1958) SA^2+SB*SC X(5058) a^2*(a*R-b*c)
X(560) a^2*(SA-SW) X(1958) a^2*SA-b^2*c^2 X(5062) a^2*(a*R+b*c)
X(590) a*R+b*c X(1959) SA^2-SB*SC X(5266) SA*r*a-S*SW
X(595) a^2*s-S*R X(1964) SA^2-SW^2 X(5277) a^3*r+b*c*S
X(604) a^2*(SA-b*c) X(1964) a^2*(SA+SW) X(5280) a*SW+S*R
X(605) a^2*(S+b*c) X(2082) SB*SC-a^2*b*c X(5299) a*SW-S*R
X(606) a^2*(S-b*c) X(2175) a^3*(a-s) X(5336) SB*SC-a^3*s
X(609) a^3+S*R X(2210) a^2*(a^2-b*c) X(5391) (b*c-S)/a