Computations for the Feuerbach inconics

intersections of tripolars

In[405]:=

iob = intersection[iso[fo], iso[fb]]

Out[405]=

{-4 (b - c)^2 c (b + c)^2 s_a s_c, 4 (a - c)^3 s_b s_o (-a c + 2 S_B), 4 a (a - b)^2 (a + b)^2 s_a s_c}

In[406]:=

ioa = intersection[iso[fo], iso[fa]]

Out[406]=

{-4 (b - c)^3 s_a s_o (b c - 2 S_A), -4 c (-a^2 + c^2)^2 s_b s_c, 4 (a - b)^2 b (a + b)^2 s_b s_c}

In[407]:=

ioc = intersection[iso[fo], iso[fc]]

Out[407]=

{4 b (b - c)^2 (b + c)^2 s_a s_b, -4 a (-a^2 + c^2)^2 s_a s_b, -4 (a - b)^3 s_c s_o (-a b + 2 S_C)}

In[408]:=

ica = intersection[iso[fc], iso[fa]]

Out[408]=

{-4 (b - c)^2 c (b + c)^2 s_b s_o, -4 (a + c)^3 s_a s_c (a c + 2 S_B), -4 a (a - b)^2 (a + b)^2 s_b s_o}

In[409]:=

icb = intersection[iso[fc], iso[fb]]

Out[409]=

{4 (b + c)^3 s_b s_c (-b c - 2 S_A), -4 c (-a^2 + c^2)^2 s_a s_o, -4 (a - b)^2 b (a + b)^2 s_a s_o}

In[410]:=

iab = intersection[iso[fa], iso[fb]]

Out[410]=

{-4 b (b - c)^2 (b + c)^2 s_c s_o, -4 a (-a^2 + c^2)^2 s_c s_o, -4 (a + b)^3 s_a s_b (a b + 2 S_C)}

perspectors from tripolar intersections

In[411]:=

intersection[avertex, ioa, bvertex, iob]

Out[411]=

{-b (b - c)^2 c (b + c)^2, -a (a - c)^2 c (a + c)^2, a (a - b)^2 b (a + b)^2}

In[414]:=

intersection[cvertex, ioc, bvertex, iob]

Out[414]=

{b (b - c)^2 c (b + c)^2, -a (a - c)^2 c (a + c)^2, -a (a - b)^2 b (a + b)^2}

In[413]:=

intersection[avertex, icb, bvertex, ica]

Out[413]=

{b (b - c)^2 c (b + c)^2, a (a - c)^2 c (a + c)^2, a (a - b)^2 b (a + b)^2}

tangent lines

In[415]:=

oatan = iso[ioa]

Out[415]=

{-1/(4 (b - c)^3 s_a s_o (b c - 2 S_A)), -1/(4 c (-a^2 + c^2)^2 s_b s_c), 1/(4 (a - b)^2 b (a + b)^2 s_b s_c)}

In[416]:=

obtan = iso[iob]

Out[416]=

{-1/(4 (b - c)^2 c (b + c)^2 s_a s_c), 1/(4 (a - c)^3 s_b s_o (-a c + 2 S_B)), 1/(4 a (a - b)^2 (a + b)^2 s_a s_c)}

In[417]:=

octan = iso[ioc]

Out[417]=

{1/(4 b (b - c)^2 (b + c)^2 s_a s_b), -1/(4 a (-a^2 + c^2)^2 s_a s_b), -1/(4 (a - b)^3 s_c s_o (-a b + 2 S_C))}

In[418]:=

cbtan = iso[icb]

Out[418]=

{1/(4 (b + c)^3 s_b s_c (-b c - 2 S_A)), -1/(4 c (-a^2 + c^2)^2 s_a s_o), -1/(4 (a - b)^2 b (a + b)^2 s_a s_o)}

In[419]:=

abtan = iso[iab]

Out[419]=

{-1/(4 b (b - c)^2 (b + c)^2 s_c s_o), -1/(4 a (-a^2 + c^2)^2 s_c s_o), -1/(4 (a + b)^3 s_a s_b (a b + 2 S_C))}

In[420]:=

catan = iso[ica]

Out[420]=

{-1/(4 (b - c)^2 c (b + c)^2 s_b s_o), -1/(4 (a + c)^3 s_a s_c (a c + 2 S_B)), -1/(4 a (a - b)^2 (a + b)^2 s_b s_o)}

pts of tangency

In[421]:=

tangency[per_, tan_] := {1/tan[[1]]^2 /per[[1]], 1/tan[[2]]^2 / per[[2]], 1/tan[[3]]^2  / per[[3]]}

In[422]:=

pob = tangency[fo, obtan]

Out[422]=

{(16 (b - c)^2 c^2 (b + c)^4 s_a^2 s_c^2)/s_oa, (16 (a - c)^4 s_b^2 s_o^2 (-a c + 2 S_B)^2)/s_ob, (16 a^2 (a - b)^2 (a + b)^4 s_a^2 s_c^2)/s_oc}

In[423]:=

poa = tangency[fo, oatan]

Out[423]=

{(16 (b - c)^4 s_a^2 s_o^2 (b c - 2 S_A)^2)/s_oa, (16 c^2 (-a^2 + c^2)^4 s_b^2 s_c^2)/((a - c)^2 s_ob), (16 (a - b)^2 b^2 (a + b)^4 s_b^2 s_c^2)/s_oc}

In[424]:=

poc = tangency[fo, octan]

Out[424]=

{(16 b^2 (b - c)^2 (b + c)^4 s_a^2 s_b^2)/s_oa, (16 a^2 (-a^2 + c^2)^4 s_a^2 s_b^2)/((a - c)^2 s_ob), (16 (a - b)^4 s_c^2 s_o^2 (-a b + 2 S_C)^2)/s_oc}

In[425]:=

pbb = tangency[fb, obtan]

Out[425]=

{-16 (b - c)^4 c^2 (b + c)^2 s_a^2 s_c, 16 (a - c)^4 s_b^2 s_o (-a c + 2 S_B)^2, -16 a^2 (a - b)^4 (a + b)^2 s_a s_c^2}

In[426]:=

paa = tangency[fa, oatan]

Out[426]=

{16 (b - c)^4 s_a^2 s_o (b c - 2 S_A)^2, -(16 c^2 (-a^2 + c^2)^4 s_b^2 s_c)/(a + c)^2, -16 (a - b)^4 b^2 (a + b)^2 s_b s_c^2}

In[427]:=

pcc = tangency[fc, octan]

Out[427]=

{-16 b^2 (b - c)^4 (b + c)^2 s_a^2 s_b, -(16 a^2 (-a^2 + c^2)^4 s_a s_b^2)/(a + c)^2, 16 (a - b)^4 s_c^2 s_o (-a b + 2 S_C)^2}

In[428]:=

pab = tangency[fa, abtan]

Out[428]=

{16 b^2 (b - c)^2 (b + c)^4 s_c^2 s_o, -(16 a^2 (-a^2 + c^2)^4 s_c s_o^2)/(a + c)^2, -16 (a + b)^4 s_a^2 s_b (a b + 2 S_C)^2}

In[429]:=

pba = tangency[fb, abtan]

Out[429]=

{-16 b^2 (b - c)^4 (b + c)^2 s_c s_o^2, (16 a^2 (-a^2 + c^2)^4 s_c^2 s_o)/(a - c)^2, -16 (a + b)^4 s_a s_b^2 (a b + 2 S_C)^2}

In[430]:=

pac = tangency[fa, catan]

Out[430]=

{16 (b - c)^2 c^2 (b + c)^4 s_b^2 s_o, -16 (a + c)^4 s_a^2 s_c (a c + 2 S_B)^2, -16 a^2 (a - b)^4 (a + b)^2 s_b s_o^2}

perspectors

In[431]:=

mmo = intersection[avertex, poa, bvertex, pob]/-4//.antirules

Out[431]=

{b^2 (b - c)^2 c^2 (b + c)^4 s_b s_c, a^2 (a - c)^2 c^2 (a + c)^4 s_a s_c, a^2 (a - b)^2 b^2 (a + b)^4 s_a s_b}

In[432]:=

mmo = intersection[cvertex, poc, bvertex, pob]/-4//.antirules

Out[432]=

{-b^2 (b - c)^2 c^2 (b + c)^4 s_b s_c, -a^2 (a - c)^2 c^2 (a + c)^4 s_a s_c, -a^2 (a - b)^2 b^2 (a + b)^4 s_a s_b}

mmo2 = intersection[avertex, paa, bvertex, pbb]/-2//.antirules

{b^2 (b - c)^4 c^2 (b + c)^2 s_a, a^2 (a - c)^4 c^2 (a + c)^2 s_b, a^2 (a - b)^4 b^2 (a + b)^2 s_c}

In[435]:=

mma = intersection[avertex, paa, bvertex, pab]/16//.antirules

Out[435]=

{(a - b)^4 b^4 (b - c)^2 (b + c)^4 s_c^3 s_o, -(a + b)^2 (a - c)^4 c^2 (a + c)^2 s_a^2 s_b^2 (a b + 2 S_C)^2, -(a - b)^4 b^2 (a + b)^4 s_a^2 s_b s_c (a b + 2 S_C)^2}

In[437]:=

mmb = intersection[avertex, pba, bvertex, pbb]/16//.antirules

Out[437]=

{-(a + b)^2 (b - c)^4 c^2 (b + c)^2 s_a^2 s_b^2 (a b + 2 S_C)^2, a^4 (a - b)^4 (a - c)^2 (a + c)^4 s_c^3 s_o, -a^2 (a - b)^4 (a + b)^4 s_a s_b^2 s_c (a b + 2 S_C)^2}

In[440]:=

intersection[avertex, mma, bvertex, mmb]/-16//.antirules

Out[440]=

{(a + b)^2 (b - c)^4 c^4 (b + c)^2 s_a s_b^3 (a b + 2 S_C)^2, a^2 (a - b)^4 b^2 (a + c)^4 s_c^4 (a c + 2 S_B)^2, a^2 (a - b)^4 (a + b)^4 c^2 s_b^3 s_c (a b + 2 S_C)^2}

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