Formulas

I am listing these to be able to look them up myself. I will get more systematic and excyclopaedic as time goes on. To some extent I am practicing formatting. The most interesting formulae are those for the circles. Notation from The Triangle Book is used.

Points

Strong

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centroid <  : 1 :  >
circumcenter <   : b^2 S_B :   >
orthocenter <   : S_CA :   >
Nine Point Center <   : b^2 S_B + 2 S_CA :   >
symmedian point <   : b^2 :   >
de Longchamps Point <   : b^2 S_B - S_CA :   >
Desmon point <   : S_B :   >
Steiner Point <   : 1/(-a^2 + c^2) :   >
Tarry Point <   : 1/(a4plus + c4plus) :   >

Weak

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incenter <  : b :  >
retro-incenter <   : 1/b :   >
Nagel Point <   : s_b :   >
Gergonne Point <   : s_ca :   >
Mittenpunkt <   : b s_b :   >
per-Mitten Point <   : b/s_b :   >
Feuerbach Point <   : (a - c)^2 s_ob :   >
per-Feuerbach Point <   : (a + c)^2 s_ca :   >

Quartile Sets

The Gergonnes

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Gergonne Point <s_bc : s_ca : s_ab>
A-Gergonne Point <s_bc : -s_ob : -s_oc>
B-Gergonne Point < -s_oa : s_ca : -s_oc>
C-Gergonne Point < -s_oa : -s_ob : s_ab>

The Nagels

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Nagel Point <s_a : s_b : s_c>
A-Nagel Point <s_o : -s_c : -s_b>
B-Nagel Point < -s_c : s_o : -s_a>
C-Nagel Point < -s_b : -s_a : s_o>

Fermats, Isodynamics

Fermats

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Fermat point <   : 1/(S_B + S_π/3) :   >
fn + fs <   : S_B/(S_B^2 - S_π/3^2) :   >
switched Fermat point <   : 1/(S_B - S_π/3) :   >
fn - fs <   : 1/(S_B^2 - S_π/3^2) :   >

Isodynamics

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Isodynamic point, In <   : b^2 (S_B + S_π/3) :   >
In + Is <   : b^2 S_B :   >
switched Isodynamic point, Is <   : b^2 (S_B - S_π/3) :   >
In - Is <   : b^2 :   >

Midpoints and infinite points

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Midpoint Fn Fs <   : (a - c)^2 (a + c)^2 :   >
Midpoint In Is <   : b^2 (-a^2 + 2 b^2 - c^2) :   >
Infinite Fn Fs <   : a^2 b^2 c^2 + 4 S_ABC - 3 b^4 S_B :   >
Infinite In Is <   : b^2 (S^2 - S_B S_Ω) :   >

Circles

Standard Strong Circles

circumcircle < 0:0:0|1 >
polarcircle < S_A : S_B : S_C | 1 >
nineptcircle < S_A : S_B : S_C | 2 >
ringcircle < S_A : S_B : S_C | 3/2 >
Brocard circle < b^2 c^2 : a^2 c^2 : a^2 b^2 | 2 S_Ω >
B-vertex circle < b^2 - c^2 : b^2 : -a^2 + b^2 | -1 >
B-edge circle < 0 : S_B : 0 | 1 >
B-median circle < S_A : 0 : S_C | 2 >
B-altitude circle < S_A^2 : 0 : S_C^2 | b^2 >
B-Apollonian circle < b^2 c^2 : 0 : -a^2 b^2 | -a^2 + c^2 >
B-orthocircle < 0 : a^2 c^2 : 0 | S_B >
B-McCay circle < b^2 : 2 S_B : b^2 | 3 >
B-Neuberg circle < b^2 : 0 : b^2 | 1 >

Families

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Schoute circle < b^2 c^2 : a^2 c^2 : a^2 b^2 | S_Ω + S Tan[ω] >
B-apex circle < b^2 : S_Ω + S Tan[ω] : b^2 | 1 >
B-centroid circle < b^2 : 2 S_B + S_Ω + S Tan[ω] : b^2 | 1 >
Tucker Circle < n/a^2 - n^2 : n/b^2 - n^2 : n/c^2 - n^2 | 1/(a^2 b^2 c^2) >

Cevian and Pedal

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Io Cevian Circle < : a c (a c + s_b^2) : | 1/2 (a + b) (a + c) (b + c) >
G Pedal Circle < : (3 a^2 - b^2 + c^2) (a^2 - b^2 + 3 c^2) : | 6 S_Ω >

Distance: Points to lines

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Io to tripolar (3 S)/(2 (abc - 8 s_abc)/(a b c)^(1/2) s_o)
Io to Io dual (S S_Ω)/(2 s_o (a b c s_o - 2 s_oabc - s_o^2 S_Ω + S_Ω^2)^(1/2))
K to a edge (a S)/(2 S_Ω)
Io to tripolar b exK ((a - c) S)/(2 a c (2 a^2 - b^2 + 2 c^2)/(a^2 c^2)^(1/2) s_o)
A to b exK tripolar S/(a^2 (2 a^2 - b^2 + 2 c^2)/(a^2 c^2)^(1/2))
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