Intersection theorem

Intersection theorem

In <a href="/en/Projective_geometry" title="Projective geometry" class="wl">projective geometry</a>, an intersection theorem or incidence theorem is a statement concerning an <a href="/en/Incidence_structure" title="Incidence structure" class="wl">incidence structure</a> – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects A and B (for instance, a point and a line). The "<a href="/en/Theorem" title="Theorem" class="wl">theorem</a>" states that, whenever a set of objects satisfies the incidences (i.e. can be identified with the objects of the incidence structure in such a way that incidence is preserved), then the objects A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't.

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For example, <a href="/en/Desargues'_theorem" title="Desargues' theorem" class="mw-redirect wl">Desargues' theorem</a> can be stated using the following incidence structure:
<ul><li>Points: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{A,B,C,a,b,c,P,Q,R,O\}}">
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 <annotation encoding="application/x-tex">{\displaystyle \{A,B,C,a,b,c,P,Q,R,O\}}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7eab15c2d55454bbbe1889fba907482b139ceb" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:27.259ex; height:2.843ex;" alt="\{A,B,C,a,b,c,P,Q,R,O\}"></li>
<li>Lines: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}}">
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 <mrow class="MJX-TeXAtom-ORD">
 <mstyle displaystyle="true" scriptlevel="0">
 <mo fence="false" stretchy="false">{</mo>
 <mi>A</mi>
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 <annotation encoding="application/x-tex">{\displaystyle \{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0380f393b9a418b2b2041fc6620a57b482290c29" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:40.737ex; height:2.843ex;" alt="\{AB,AC,BC,ab,ac,bc,Aa,Bb,Cc,PQ\}"></li>
<li>Incidences (in addition to obvious ones such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A,AB)}">
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 <annotation encoding="application/x-tex">{\displaystyle (A,AB)}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/130273e9ab92a96d8251e7149855e504cfab261c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.093ex; height:2.843ex;" alt="(A,AB)">): <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}}">
 <semantics>
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 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
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 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
 <mi>O</mi>
 <mo>,</mo>
 <mi>C</mi>
 <mi>c</mi>
 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
 <mi>P</mi>
 <mo>,</mo>
 <mi>B</mi>
 <mi>C</mi>
 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
 <mi>P</mi>
 <mo>,</mo>
 <mi>b</mi>
 <mi>c</mi>
 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
 <mi>Q</mi>
 <mo>,</mo>
 <mi>A</mi>
 <mi>C</mi>
 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
 <mi>Q</mi>
 <mo>,</mo>
 <mi>a</mi>
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 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
 <mi>R</mi>
 <mo>,</mo>
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 <mo stretchy="false">)</mo>
 <mo>,</mo>
 <mo stretchy="false">(</mo>
 <mi>R</mi>
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 <annotation encoding="application/x-tex">{\displaystyle \{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/940187472d738388ca8ba18ff7701aef51de9579" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:77.724ex; height:2.843ex;" alt="\{(O,Aa),(O,Bb),(O,Cc),(P,BC),(P,bc),(Q,AC),(Q,ac),(R,AB),(R,ab)\}"></li></ul>
The implication is then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (R,PQ)}">
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 <mstyle displaystyle="true" scriptlevel="0">
 <mo stretchy="false">(</mo>
 <mi>R</mi>
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 <annotation encoding="application/x-tex">{\displaystyle (R,PQ)}</annotation>
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</math><img loading="lazy"src="https://wikimedia.org/api/rest_v1/media/math/render/svg/459f8a4216af3a813e98f280cbe0ceaa1dd6a11c" class="mwe-math-fallback-image-inline" aria-hidden="true" style="vertical-align: -0.838ex; width:8.191ex; height:2.843ex;" alt="(R,PQ)">—that point R is incident with line PQ.

In projective geometry, an intersection theorem or incidence theorem is a statement concerning an incidence structure – consisting of points, lines, and possibly higher-dimensional objects and their incidences – together with a pair of objects A and B . The "theorem" states that, whenever a set of objects satisfies the incidences , then the objects A and B must also be incident. An intersection theorem is not necessarily true in all projective geometries; it is a property that some geometries satisfy but others don't. For example, Desargues' theorem can be stated using the following incidence structure: The implication is then {\displaystyle }
  
—that point R is incident with line PQ. 