Computations for the Mandart inconics

perspectors: the Nagel points

{no, na, nb, nc}

{{s_a, s_b, s_c}, {s_o, -s_c, -s_b}, {-s_c, s_o, -s_a}, {-s_b, -s_a, s_o}}

centers: the mittenpunkts

{mo, ma, mb, mc}

{{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}}

intersections of tripolars

The tripolars of the perspectors meet at six points. iob is the intersection of the tripolars of M_o and M_b.

iob = intersection[go, gb]/4

{-a s_a s_c, (a - c) s_b s_o, c s_a s_c}

ioa = intersection[go, ga]/4

{-(b - c) s_a s_o, b s_b s_c, -c s_b s_c}

ioc = intersection[go, gc]/4

{a s_a s_b, -b s_a s_b, -(a - b) s_c s_o}

ica = intersection[gc, ga]/4

{a s_b s_o, -(a + c) s_a s_c, c s_b s_o}

icb = intersection[gc, gb]/4

{(b + c) s_b s_c, -b s_a s_o, -c s_a s_o}

iab = intersection[ga, gb]/4

{a s_c s_o, b s_c s_o, -(a + b) s_a s_b}

The tripolars intersect at points related to the incenter.

The first two are harmonic associates of the incenter.

intersection[avertex, ioa, bvertex, iob]

{-a, -b, c}

intersection[cvertex, ioc, bvertex, iob]

{-a, b, c}

intersection[cvertex, iab, bvertex, ica]

{a, b, c}

        

tangent lines

The tripolars of the intersections are the missing tangent lines, each to two conics. There are six of these lines. oatan is the tangent to the o and a conics that is not an edge of ABC.

oatan = iso[ioa/4]

{-4/((b - c) s_a s_o), 4/(b s_b s_c), -4/(c s_b s_c)}

obtan = iso[iob/4]

{-4/(a s_a s_c), 4/((a - c) s_b s_o), 4/(c s_a s_c)}

octan = iso[ioc/4]

{4/(a s_a s_b), -4/(b s_a s_b), -4/((a - b) s_c s_o)}

cbtan = iso[icb/4]

{4/((b + c) s_b s_c), -4/(b s_a s_o), -4/(c s_a s_o)}

abtan = iso[iab]

{1/(a s_c s_o), 1/(b s_c s_o), -1/((a + b) s_a s_b)}

catan = iso[ica/4]

{4/(a s_b s_o), -4/((a + c) s_a s_c), 4/(c s_b s_o)}

The tangents intersect at points related to the isotomic incenter.

The first two are harmonic associates of the isotomic incenter.

intersection[avertex, oatan, bvertex, obtan]

{-b c, -a c, a b}

intersection[cvertex, octan, bvertex, obtan]

{b c, -a c, -a b}

intersection[cvertex, abtan, bvertex, catan]

{-b c, -a c, -a b}

        

pts of tangency

Each of the six tangent lines has two point of tangency, making three for each of the four inconics. pob is the b point on the o circle.

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

pob = tangency[no, obtan] 16

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

poa = tangency[no, oatan] 16

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

poc = tangency[no, octan] 16

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

pbb = tangency[nb, obtan] 16

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

paa = tangency[na, oatan] 16

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

pcc = tangency[nc, octan] 16

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

pab = tangency[na, abtan]

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

pba = tangency[nb, abtan]

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

pac = tangency[na, catan] 16

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

perspectors

The points of tangency are in perspective at the proGergonne and proNagel points.

proGo

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

{a^2 s_b s_c, b^2 s_a s_c, c^2 s_a s_b}

pro No

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

{a^2 s_a, b^2 s_b, c^2 s_c}

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