computations for the Mittenpunkt inconics

perspectors: the mittenpunkt points

In[497]:=

{mo, ma, mb, mc}

Out[497]=

{{a s_a, b s_b, c s_c}, {a s_o, b s_c, c s_b}, {a s_c, b s_o, c s_a}, {a s_b, b s_a, c s_o}}

centers:

In[502]:=

(-subordinate[iso[mo]]//.rules3//Expand//Factor)//.antirules

Out[502]=

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

The tripolars of the perspectors meet at six points

In[500]:=

-a b + b^2 - a c - 2 b c + c^2//asymmetrize

Out[500]=

-2 a s_a - 4 s_bc

a-coordinate of center

(as_a + 2s_bc)/(bcs_bs_c)    ~        a s_a (as_a + 2s_bc)

intersections of tripolars

The tripolars of the four perspectors meet at six points.

In[441]:=

iob = intersection[iso[mo], iso[mb]]/8

Out[441]=

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

In[442]:=

ioa = intersection[iso[mo], iso[ma]]/8

Out[442]=

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

In[443]:=

ioc = intersection[iso[mo], iso[mc]]/8

Out[443]=

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

In[444]:=

ica = intersection[iso[mc], iso[ma]]/8

Out[444]=

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

In[445]:=

icb = intersection[iso[mc], iso[mb]]/8

Out[445]=

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

In[446]:=

iab = intersection[iso[ma], iso[mb]]/8

Out[446]=

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

perspectors from tripolar intersections

The perspectors are perspective at points related to the Clawson points.

In[447]:=

intersection[avertex, ioa, bvertex, iob]

Out[447]=

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

In[448]:=

intersection[cvertex, ioc, bvertex, iob]

Out[448]=

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

In[449]:=

intersection[avertex, icb, bvertex, ica]

Out[449]=

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

tangent lines

The tripolars of the intersections are each tangent to two conics.

oatan = iso[ioa]

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

obtan = iso[iob]

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

octan = iso[ioc]

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

cbtan = iso[icb]

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

abtan = iso[iab]

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

catan = iso[ica]

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

intersection perspectors

The tangent lines produce these perspectors, which are points I have never heard of.

mta = intersection[catan, abtan]/-4

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

mtc = intersection[cbtan, catan]/-4

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

intersection[avertex, mta, cvertex, mtc]/-4//.antirules

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

pts of tangency

The tripolars of the intersections are the missing tangent lines, each to two conics. There are six of these lines

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

pob = tangency[mo, obtan]

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

poa = tangency[mo, oatan]

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

poc = tangency[mo, octan]

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

pbb = tangency[mb, obtan]

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

paa = tangency[ma, oatan]

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

pcc = tangency[mc, octan]

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

pab = tangency[ma, abtan]

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

pba = tangency[mb, abtan]

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

pac = tangency[ma, catan]

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

pca = tangency[mc, catan]

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

pbc = tangency[mb, cbtan]

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

pcb = tangency[mc, cbtan]

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

perspectors

do = intersection[avertex, poa, bvertex, pob]/-64/.antirules

{a s_b s_c S_B^2 S_C^2, b s_a s_c S_A^2 S_C^2, c s_a s_b S_A^2 S_B^2}

da = intersection[avertex, paa, bvertex, pab]/256/.antirules

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

db = intersection[avertex, pba, bvertex, pbb]/-256/.antirules

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

qo =   intersection[avertex, da, bvertex, db]/32/.antirules

{a s_a S_B^2 S_C^2, b s_b S_A^2 S_C^2, c s_c S_A^2 S_B^2}

mdo = intersection[avertex, paa, bvertex, pbb]/-32//.antirules

{a s_a S_B^2 S_C^2, b s_b S_A^2 S_C^2, c s_c S_A^2 S_B^2}

mdc = intersection[avertex, paa, bvertex, pab]/256/.antirules

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

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