Dual of Circumcircle
[From Peter Moses]
Dual circumcircle, center 141 mK, perspector 76 (R = tK)
{338 {(b2 – c2)2 / a2}, 1086 wmS}
The 1086x are the four meets with the Steiner inellipse.
This conic and the Steiner inellipse have the dual of the Steiner point as 4th tangent.
These points are generated by the natural mappings (in grey). ETC numbers are in red and the y coordinate is given in black. The colors in this chart show point originating from the various triangle points.
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line at infinity
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Steiner dual
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Steiner inellipse
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R-inconic
dual circumconic |
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( : y : )
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( : (c2–a2) y : )
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( : y2 : )
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( : y2/a2 : )
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From the Incenter
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514 c–a
∞•~Io |
? (c2–a2)(c–a)
- |
1086 (c–a)2
weakened KH center |
? (c–a)2/a2
- |
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519 c+a–2b
∞•(G—Io) |
? (c2–a2)(c+a–2b)
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(c+a–2b)2
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? (c+a–2b)2/b2
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513 b(c–a)
twS = ∞•~tIo |
? (c2–a2)b(c–a)
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b2(c–a)2
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1086 (c–a)2
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900 (c-a)(c+a-2b)
(∞•~190o) |
? (c2–a2)(c-a)(c+a-2b)
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(c-a)2(c+a-2b)2
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? (c–a)2(c+a–2b)2/b2
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812 (c–a)(b2–ca)
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? (c2–a2)(c–a)(b2–ca)
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? (c–a)2(b2–ca)2
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? (c–a)2(b2–ca)2/b2
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? (c+a)(b2–ca)
∞•(Io—tIo) |
? (c2–a2) (c+a)(b2–ca)
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? (c+a)2(b2–ca)2
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? (c+a)2(b2–ca)2/b2
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? b(c2+a2–ab–bc)
∞•(Go—No) |
? (c2–a2)b(c2+a2–ab–bc)
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? b2(c2+a2–ab–bc)2
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? (c2+a2–ab–bc)2
(tT)2 |
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? b(c-a)sbb,
∞•~Go2 |
? (c2–a2)b(c-a)sbb,
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? b2(c-a)2sbbbb,
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? (c-a)2sbbbb,
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From the Gergonne point
also the Mittenpunkt |
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g109 (c–a)sb
∞•~Go |
? (c2–a2)(c–a)sb
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(c–a)2sb2
ABCGNo center |
? (c–a)2 sb2/b2
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? csc+asa–2bsb
∞•(G—Go) |
? (c2–a2)(csc+asa–2bsb)
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(csc+asa–2bsb)2
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(csc+asa–2bsb)/b2
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From the Symmedian point
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523 c2–a2
tS = ∞•~K |
115 (c2–a2)2
KH center |
115 (c2-a2)2
KH center ~S intersections |
338 (c2–a2)2/b2
intersection with MacBeath inconic |
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524 c2+a2–2b2
∞•(G—K) |
690 (c2–a2)(c2+a2–2b2)
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(c2+a2-2b2)2
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(c2–a2)2(b4–c2a2)2/b2
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(c2–a2)(c2+a2–2b2)
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1648 (c2–a2)2(c2+a2–2b2)
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512 b2(c2-a2)
gS = ∞•~tK |
? b2(c2-a2)2
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b4(c2–a2)2
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? b2(c2–a2)2
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g98 b2SB2-
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? (c2-a2)b2SB2-
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b4(SB2-)2
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? b2(SB2–)2
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From the Orthocenter
also the Circumcenter |
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? (c2–a2)SB
∞•~H = ∞•~O |
125 (c2–a2)2SB
J, ~S intersection |
(c2-a2)2 SB2
Eulerian center |
125 (c2–a2)2SB2/b2
J, ~S intersection |
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30 SBC+SAB–2SCA
= (b2 SA – 2 SBC) ∞•(G—H) infinite point on Euler line |
? (c2–a2)(SBC+SAB–2SCA)
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(SBC+SAB–2SCA)2
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(SBC+SAB–2SCA)2/b2
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? b2(c2–a2)SBB
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? b2(c2–a2)2SBB
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b4(c2–a2)2
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? b2(c2–a2)2SBBB
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Fissile points
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