# proof of Desargues’ theorem

The claim is that if triangles $ABC$ and $XYZ$ are perspective from a point $P$, then they are perspective from a line (meaning that the three points

$$AB\cdot XY\mathit{\hspace{1em}\hspace{1em}}BC\cdot YZ\mathit{\hspace{1em}\hspace{1em}}CA\cdot ZX$$ |

are collinear^{}) and conversely.

Since no three of $A,B,C,P$ are collinear, we can lay down
homogeneous coordinates^{} such that

$$A=(1,0,0)\mathit{\hspace{1em}\hspace{1em}}B=(0,1,0)\mathit{\hspace{1em}\hspace{1em}}C=(0,0,1)\mathit{\hspace{1em}\hspace{1em}}P=(1,1,1)$$ |

By hypothesis, there are scalars $p,q,r$ such that

$$X=(1,p,p)\mathit{\hspace{1em}\hspace{1em}}Y=(q,1,q)\mathit{\hspace{1em}\hspace{1em}}Z=(r,r,1)$$ |

The equation for a line through $({x}_{1},{y}_{1},{z}_{1})$ and $({x}_{2},{y}_{2},{z}_{2})$ is

$$({y}_{1}{z}_{2}-{z}_{1}{y}_{2})x+({z}_{1}{x}_{2}-{x}_{1}{z}_{2})y+({x}_{1}{y}_{2}-{y}_{1}{x}_{2})z=0,$$ |

giving us equations for six lines:

$AB$ | $:$ | $z=0$ | ||

$BC$ | $:$ | $x=0$ | ||

$CA$ | $:$ | $y=0$ | ||

$XY$ | $:$ | $(pq-p)x+(pq-q)y+(1-pq)z=0$ | ||

$YZ$ | $:$ | $(1-qr)x+(qr-q)y+(qr-r)z=0$ | ||

$ZX$ | $:$ | $(rp-p)x+(1-rp)y+(rp-r)z=0$ |

whence

$AB\cdot XY$ | $=$ | $(pq-q,-pq+p,0)$ | ||

$BC\cdot YZ$ | $=$ | $(0,qr-r,-qr+q)$ | ||

$CA\cdot ZX$ | $=$ | $(-rp+r,0,rp-p).$ |

As claimed, these three points are collinear, since the determinant

$$\left|\begin{array}{ccc}\hfill pq-q\hfill & \hfill -pq+p\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill qr-r\hfill & \hfill -qr+q\hfill \\ \hfill -rp+r\hfill & \hfill 0\hfill & \hfill rp-p\hfill \end{array}\right|$$ |

is zero. (More precisely, all three points are on the line

$$p(q-1)(r-1)x+(p-1)q(r-1)y+(p-1)(q-1)rz=0.)$$ |

Since the hypotheses are self-dual, the converse is true also, by the principle of duality.

Title | proof of Desargues’ theorem |
---|---|

Canonical name | ProofOfDesarguesTheorem |

Date of creation | 2013-03-22 13:47:51 |

Last modified on | 2013-03-22 13:47:51 |

Owner | drini (3) |

Last modified by | drini (3) |

Numerical id | 5 |

Author | drini (3) |

Entry type | Proof |

Classification | msc 51A30 |