Points on the Steiner ellipse and the Circumcircle
These points project down from the line at infinity. The isotomic conjugate of a point at infinity is on the Steiner ellipse. The isogonal conjugate of a point at infinity is on the circumcircle. The "pro" operation takes the Steiner ellipse to the circumcircle. The line between corresponding points goes through S, the Steiner point, which is the fourth intersection of the two conics.
Figure: This shows corresponding pairs of points on the Steiner ellipse and the circumcircle. Strong points are bold; others are shown red or gree with all their versions. The line between corresponding pairs goes through S, the Steiner point. The bold red lines are the Steiner axes and I am not sure why they are in the picture. The operation that takes the Steiner ellipse to the circumcircle is the "pro" operation. Projection from the Steiner point does this for points on the Steiner ellipse, shown as the lines emanating from S.
A Sketchpad document has many of these points for your entertainment. Movie soon.
This table gives the ETC number in red, the y coordinate in black and comments in green. Each point on an object can be regarded as orginating from a point in the triangle plane. Points of origin are indicated by the colored regions.
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line at infinity
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Steiner ellipse
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Circumcircle
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From the Incenter
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514 c–a
∞•~Io |
190 1/(c–a)
t(∞•~Io) |
101 b2/(c–a)
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519 c+a–2b
∞•(G—Io) |
903 1/(c+a–2b)
t(∞•(G—Io)) |
106 b2/(c+a–2b)
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513 b(c–a)
twS = ∞•~tIo |
668 1/b(c–a)
t(∞•~tIo) |
100 b/(c–a)
wF = focus of Yff parabola |
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900 (c-a)(c+a-2b)
(∞•~190o) |
t900 1/(c-a)(c+a-2b)
t(∞•~190o) |
901 b2/(c-a)(c+a–2b)
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812 (c–a)(b2–ca)
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? 1/(c–a)(b2–ca)
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813 b2/(c–a)(b2–ca)
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? (c+a)(b2–ca)
∞•(Io—tIo) |
? 1/(c+a)(b2–ca)
t(∞•(Io—tIo)) |
? b2/(c+a)(b2–ca)
g(∞•(Io—tIo)) |
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? b(c2+a2–ab–bc)
∞•(Go—tNo) |
2481 1/b(c2+a2–ab–bc)
4th of SE and Feuerbach |
105 b/(c2+a2-ab-bc)
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? b(c-a)sbb,
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? 1/b(c-a)sbb,
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? b/(c-a)sbb,
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? (c–a)sb(b2–ca)
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? 1/(c–a)sb(b2–ca)
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? b2/(c–a)sb(b2–ca)
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? (c–a)(b2+bc+ab+2ca)
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? 1/(c–a)(b2+bc+ab+2ca)
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? b2/(c–a)(b2+bc+ab+2ca)
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? (c–a)(b2+bc+ab+ca)
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? 1/ (c–a)(b2+bc+ab+ca)
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? b2/(c–a)(b2+bc+ab+ca)
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? (c–a)(c2+a2+ca)
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? 1/c–a)(c2+a2+ca)
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? b2/(c–a)(c2+a2+ca)
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? (c–a)(c2+a2–bc–ab–2ca)
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? 1/(c–a)(c2+a2–bc–ab–2ca)
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105 b2/(c–a)(c2+a2–bc–ab–2ca)
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? b2(c–a)sb(c2+a2–ab–bc)
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? 1/b2(c–a)sb(c2+a2–ab–bc)
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1/(c–a)sb(c2+a2–ab–bc)
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From the Gergonne point
also the Mittenpunkt |
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g109 (c–a)sb
∞•~Go |
r109 1/(c–a)sb
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109 b2/(c–a)sb
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? csc+asa–2bsb
∞•(G—Go) |
? 1/(csc+asa–2bsb)
t(∞•(G—Go)) |
? b2/(csc+asa–2bsb)
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From the Symmedian point
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523 c2–a2
tS = ∞•~K |
99 1/c2–a2
t(∞•~K) the Steiner point |
110 b2/(c2–a2)
the focus of the Kiepert Parabola |
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524 c2+a2–2b2
∞•(G—K) |
671 1/(c2+a2–2b2)
t(∞•(G—K)) |
111 b2/(c2+a2–2b2)
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512 b2(c2-a2)
gS = ∞•~tK |
670 1/b2(c2-a2)
rS = t(∞•~tK) |
99 1/(c2–a2)
Steiner point |
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g98 b2SB2-
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r98 1/b2SB2-
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98 1/SB2-
Tarry point |
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From the Orthocenter
also the Circumcenter |
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? (c2–a2)SB
∞•~H = ∞•~O |
? 1/(c2–a2)SB
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? b2/(c2–a2)SB
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30 SBC+SAB–2SCA
= (b2 SA – 2 SBC) ∞•(G—H) infinite point on Euler line |
1494 1/(SBC+SAB–2SCA)
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74 b2/(SBC+SAB–2SCA)
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? b2(c2–a2)SBB
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? 1/b2(c2–a2)SBB
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? 1/(c2–a2)SBB
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Fissile points
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r1113, r1114 b2(J–1)SB+2bSCA
D-tO meets Steiner ellipse |
1113,1114 b (J–1) SB + 2 SCA Euler meets circumcircle
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not so sure what to make of these
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? b(c-a)sb(4SBB-c2a2)
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? 1/b(c-a)sb(4SBB-c2a2)
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b/(c-a)sb(4SBB-c2a2)
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