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Desargues's theorem
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===Three-dimensional proof=== Desargues's theorem is true for any projective space of dimension at least 3, and more generally for any projective space that can be embedded in a space of dimension at least 3. Desargues's theorem can be stated as follows: :If lines {{math|{{overline|''Aa''}}, {{overline|''Bb''}}}} and {{math|{{overline|''Cc''}}}} are concurrent (meet at a point), then :the points {{math|{{overline|''AB''}} β© {{overline|''ab''}}, {{overline|''AC''}} β© {{overline|''ac''}}}} and {{math|{{overline|''BC''}} β© {{overline|''bc''}}}} are [[collinear]]. The points {{math|''A'', ''B'', ''a''}} and {{math|''b''}} are coplanar (lie in the same plane) because of the assumed concurrency of {{math|{{overline|''Aa''}}}} and {{math|{{overline|''Bb''}}}}. Therefore, the lines {{math|{{overline|''AB''}}}} and {{math|{{overline|''ab''}}}} belong to the same plane and must intersect. Further, if the two triangles lie on different planes, then the point {{math|{{overline|''AB''}} β© {{overline|''ab''}}}} belongs to both planes. By a symmetric argument, the points {{math|{{overline|''AC''}} β© {{overline|''ac''}}}} and {{math|{{overline|''BC''}} β© {{overline|''bc''}}}} also exist and belong to the planes of both triangles. Since these two planes intersect in more than one point, their intersection is a line that contains all three points. This proves Desargues's theorem if the two triangles are not contained in the same plane. If they are in the same plane, Desargues's theorem can be proved by choosing a point not in the plane, using this to lift the triangles out of the plane so that the argument above works, and then projecting back into the plane. The last step of the proof fails if the projective space has dimension less than 3, as in this case it is not possible to find a point not in the plane. [[Monge's theorem]] also asserts that three points lie on a line, and has a proof using the same idea of considering it in three rather than two dimensions and writing the line as an intersection of two planes.
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