Theorem on the collinearity of three points generated from a hexagon inscribed on a conic

Pascal&#039;s theorem

Pascal's theorem

In <a href="/en/Projective_geometry" title="Projective geometry" class="wl">projective geometry</a>, Pascal's theorem (also known as the hexagrammum mysticum theorem, <a href="/en/Latin" title="Latin" class="wl">Latin</a> for mystical <a href="/en/Hexagram" title="Hexagram" class="wl">hexagram</a>) states that if six arbitrary points are chosen on a <a href="/en/Conic_section" title="Conic section" class="wl">conic</a> (which may be an <a href="/en/Ellipse" title="Ellipse" class="wl">ellipse</a>, <a href="/en/Parabola" title="Parabola" class="wl">parabola</a> or <a href="/en/Hyperbola" title="Hyperbola" class="wl">hyperbola</a> in an appropriate <a href="/en/Affine_plane" title="Affine plane" class="wl">affine plane</a>) and joined by line segments in any order to form a <a href="/en/Hexagon" title="Hexagon" class="wl">hexagon</a>, then the three pairs of opposite <a href="/en/Edge_(geometry)" title="Edge (geometry)" class="wl">sides</a> of the hexagon (<a href="/en/Extended_side" title="Extended side" class="wl">extended</a> if necessary) meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after <a href="/en/Blaise_Pascal" title="Blaise Pascal" class="wl">Blaise Pascal</a>.

<div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none" about="#mwt1">Theorem on the collinearity of three points generated from a hexagon inscribed on a conic</div>
<figure class="wiki-thumb"><a href="/en/File:Pascaltheoremgenericwithlabels.svg" class="mw-file-description image media-thumb" data-hash="File:Pascaltheoremgenericwithlabels.svg"><img loading="lazy" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Pascaltheoremgenericwithlabels.svg/250px-Pascaltheoremgenericwithlabels.svg.png" decoding="async" data-file-width="500" data-file-height="500" data-file-type="drawing" height="250" width="250" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Pascaltheoremgenericwithlabels.svg/375px-Pascaltheoremgenericwithlabels.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Pascaltheoremgenericwithlabels.svg/500px-Pascaltheoremgenericwithlabels.svg.png 2x" alt="Pascaltheoremgenericwithlabels.svg"></a><figcaption>Pascal line GHK of self-crossing hexagon ABCDEF inscribed in ellipse. Opposite sides of hexagon have the same color.</figcaption></figure>
<figure class="wiki-thumb"><a href="/en/File:Pascal'sTheoremLetteredColored.PNG" class="mw-file-description image media-thumb" data-hash="File:Pascal'sTheoremLetteredColored.PNG"><img loading="lazy" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Pascal%27sTheoremLetteredColored.PNG/250px-Pascal%27sTheoremLetteredColored.PNG" decoding="async" data-file-width="650" data-file-height="550" data-file-type="bitmap" height="212" width="250" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Pascal%27sTheoremLetteredColored.PNG/375px-Pascal%27sTheoremLetteredColored.PNG 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Pascal%27sTheoremLetteredColored.PNG/500px-Pascal%27sTheoremLetteredColored.PNG 2x" alt="Pascal%27sTheoremLetteredColored.PNG"></a><figcaption>Self-crossing hexagon ABCDEF, inscribed in a circle. Its sides are extended so that pairs of opposite sides intersect on Pascal's line. Each pair of extended opposite sides has its own color: one red, one yellow, one blue. Pascal's line is shown in white.</figcaption></figure>

The theorem is also valid in the <a href="/en/Euclidean_plane" title="Euclidean plane" class="wl">Euclidean plane</a>, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel.

This theorem is a generalization of <a href="/en/Pappus's_hexagon_theorem" title="Pappus's hexagon theorem" class="wl">Pappus's (hexagon) theorem</a>, which is the special case of a <a href="/en/Degenerate_conic" title="Degenerate conic" class="wl">degenerate conic</a> of two lines with three points on each line.

In projective geometry, Pascal's theorem states that if six arbitrary points are chosen on a conic and joined by line segments in any order to form a hexagon, then the three pairs of opposite sides of the hexagon meet at three points which lie on a straight line, called the Pascal line of the hexagon. It is named after Blaise Pascal. The theorem is also valid in the Euclidean plane, but the statement needs to be adjusted to deal with the special cases when opposite sides are parallel. This theorem is a generalization of Pappus's theorem, which is the special case of a degenerate conic of two lines with three points on each line.  