Dualize this Projective Geometry Statement The Next CEO of Stack OverflowWhitehead's axioms of projective geometryMethod for Visualizing Projective SpaceProjective Geometry True/False QuestionsCollinearity of points in a projective settingTwo triangles cirumcribed a conic problemHow do you transform from affine space to projective space?Two Parallel Lines Give Rise to a Third Parallel NaturallyResults of projective and Euclidean geometryProjective geometry and planes equationsProjective Plane of order n
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Dualize this Projective Geometry Statement
The Next CEO of Stack OverflowWhitehead's axioms of projective geometryMethod for Visualizing Projective SpaceProjective Geometry True/False QuestionsCollinearity of points in a projective settingTwo triangles cirumcribed a conic problemHow do you transform from affine space to projective space?Two Parallel Lines Give Rise to a Third Parallel NaturallyResults of projective and Euclidean geometryProjective geometry and planes equationsProjective Plane of order n
$begingroup$
In the Extended Euclidean Plane, let l and m be two lines that intersect at the point ). Let A, B, C be three points on l other than o and let A', B', C' be three points on m other than o. Assume that the line AA', BB', and CC' are concurrent at a point P. Let E= AB'(intersect)A'B, F=AC'(intersect)A'C. Then the points O,E,F are collinear.
Here's what I wrote, but idk if it's correct.
In the Extended Euclidean Plane, let L and M be two points lying on the line O. Let a,b,c be three lines containing L other than line O and let a'b'c' be three lines containing M other than line O. Assume that the points A and A' are collinear, B and B' are collinear, and C and C' are collinear and the lines AA', BB', and CC' all contain P. Let line e contains the points A',B,A,B' and line f contain the points A,C',A',C. Then, the lines o,e, and f are concurrent.
projective-geometry
asked 2 days ago
lj_growllj_growl
557
$endgroup$
add a comment |
$begingroup$
In the Extended Euclidean Plane, let l and m be two lines that intersect at the point ). Let A, B, C be three points on l other than o and let A', B', C' be three points on m other than o. Assume that the line AA', BB', and CC' are concurrent at a point P. Let E= AB'(intersect)A'B, F=AC'(intersect)A'C. Then the points O,E,F are collinear.
Here's what I wrote, but idk if it's correct.
In the Extended Euclidean Plane, let L and M be two points lying on the line O. Let a,b,c be three lines containing L other than line O and let a'b'c' be three lines containing M other than line O. Assume that the points A and A' are collinear, B and B' are collinear, and C and C' are collinear and the lines AA', BB', and CC' all contain P. Let line e contains the points A',B,A,B' and line f contain the points A,C',A',C. Then, the lines o,e, and f are concurrent.
projective-geometry
asked 2 days ago
lj_growllj_growl
557
$endgroup$
$begingroup$
When you say "assume that the points $A$ and $A'$ are collinear," it seems that these points have not been defined yet in the dual statement.
$endgroup$
– hunter
2 days ago
$begingroup$
@hunter I see. What exactly are these points though? That's where I'm not understanding how my dual statement works.
$endgroup$
– lj_growl
2 days ago
$begingroup$
The dual to the line $AA'$ will be the intersection of the duals of $A$ and $A'$.
$endgroup$
– hunter
2 days ago
add a comment |
$begingroup$
In the Extended Euclidean Plane, let l and m be two lines that intersect at the point ). Let A, B, C be three points on l other than o and let A', B', C' be three points on m other than o. Assume that the line AA', BB', and CC' are concurrent at a point P. Let E= AB'(intersect)A'B, F=AC'(intersect)A'C. Then the points O,E,F are collinear.
Here's what I wrote, but idk if it's correct.
In the Extended Euclidean Plane, let L and M be two points lying on the line O. Let a,b,c be three lines containing L other than line O and let a'b'c' be three lines containing M other than line O. Assume that the points A and A' are collinear, B and B' are collinear, and C and C' are collinear and the lines AA', BB', and CC' all contain P. Let line e contains the points A',B,A,B' and line f contain the points A,C',A',C. Then, the lines o,e, and f are concurrent.
projective-geometry
asked 2 days ago
lj_growllj_growl
557
$endgroup$
In the Extended Euclidean Plane, let l and m be two lines that intersect at the point ). Let A, B, C be three points on l other than o and let A', B', C' be three points on m other than o. Assume that the line AA', BB', and CC' are concurrent at a point P. Let E= AB'(intersect)A'B, F=AC'(intersect)A'C. Then the points O,E,F are collinear.
Here's what I wrote, but idk if it's correct.
In the Extended Euclidean Plane, let L and M be two points lying on the line O. Let a,b,c be three lines containing L other than line O and let a'b'c' be three lines containing M other than line O. Assume that the points A and A' are collinear, B and B' are collinear, and C and C' are collinear and the lines AA', BB', and CC' all contain P. Let line e contains the points A',B,A,B' and line f contain the points A,C',A',C. Then, the lines o,e, and f are concurrent.
projective-geometry
projective-geometry
asked 2 days ago
lj_growllj_growl
557
asked 2 days ago
lj_growllj_growl
557
edited 2 days ago
lj_growl
asked 2 days ago
lj_growllj_growl
557
asked 2 days ago
lj_growllj_growl
557
asked 2 days ago
lj_growllj_growl
557
557
$begingroup$
When you say "assume that the points $A$ and $A'$ are collinear," it seems that these points have not been defined yet in the dual statement.
$endgroup$
– hunter
2 days ago
$begingroup$
@hunter I see. What exactly are these points though? That's where I'm not understanding how my dual statement works.
$endgroup$
– lj_growl
2 days ago
$begingroup$
The dual to the line $AA'$ will be the intersection of the duals of $A$ and $A'$.
$endgroup$
– hunter
2 days ago
add a comment |
$begingroup$
When you say "assume that the points $A$ and $A'$ are collinear," it seems that these points have not been defined yet in the dual statement.
$endgroup$
– hunter
2 days ago
$begingroup$
@hunter I see. What exactly are these points though? That's where I'm not understanding how my dual statement works.
$endgroup$
– lj_growl
2 days ago
$begingroup$
The dual to the line $AA'$ will be the intersection of the duals of $A$ and $A'$.
$endgroup$
– hunter
2 days ago
$begingroup$
When you say "assume that the points $A$ and $A'$ are collinear," it seems that these points have not been defined yet in the dual statement.
$endgroup$
– hunter
2 days ago
$begingroup$
When you say "assume that the points $A$ and $A'$ are collinear," it seems that these points have not been defined yet in the dual statement.
$endgroup$
– hunter
2 days ago
$begingroup$
@hunter I see. What exactly are these points though? That's where I'm not understanding how my dual statement works.
$endgroup$
– lj_growl
2 days ago
$begingroup$
@hunter I see. What exactly are these points though? That's where I'm not understanding how my dual statement works.
$endgroup$
– lj_growl
2 days ago
$begingroup$
The dual to the line $AA'$ will be the intersection of the duals of $A$ and $A'$.
$endgroup$
– hunter
2 days ago
$begingroup$
The dual to the line $AA'$ will be the intersection of the duals of $A$ and $A'$.
$endgroup$
– hunter
2 days ago
add a comment |
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$begingroup$
When you say "assume that the points $A$ and $A'$ are collinear," it seems that these points have not been defined yet in the dual statement.
$endgroup$
– hunter
2 days ago
$begingroup$
@hunter I see. What exactly are these points though? That's where I'm not understanding how my dual statement works.
$endgroup$
– lj_growl
2 days ago
$begingroup$
The dual to the line $AA'$ will be the intersection of the duals of $A$ and $A'$.
$endgroup$
– hunter
2 days ago