# Desargues' Theorem

Let A_{1}B_{1}C_{1} and A_{2}B_{2}C_{2} be two triangles. Consider two conditions:

- Lines A
_{1}A_{2}, B_{2}B_{2}, C_{1}C_{2}joining the corresponding vertices are concurrent. - Points ab, bc, ca of intersection of the (extended) sides A
_{1}B_{1}and A_{2}B_{2}, B_{1}C_{1}and B_{2}C_{2}, C_{1}A_{1}and C_{2}A_{2}, respectively, are collinear.

Desargues' Theorem claims that 1. implies 2. It's dual asserts that 1. follows from 2. In particular, the dual to Desragues' theorem coincides with its converse.

(In the applet below each of the triangles as well as each of the vertices is draggable.)

Two triangles that satisfy the first condition are said to be *perspecitve from a point*. Two triangles that satisfy the second condition are said to be *perspecitve from a line*. Desargues' theorem thus claims that two triangles perspective from a point are perspective from a line. Its dual asserts that two triangles perspective from a line are also perspective from a point.

Monge's theorem can be derived from that of Desargues and in fact is the latter in disguise. The existence of an orthic axis of a triangle is also an immediate consequence of Desargues' Theorem. In a somewhat disguised form Desargues' Theorem establishes a relationship between a triangle and a cevian triangle of a point not on a triangle itself.

Curiously, Desragues' theorem admits an intuitive proof if considered as a statement in the 3-dimensional space, but is not as easy in the 2-dimensional case, where it is often taken as an axiom.

Following is the proof (kindly supplied by Hubert Shutrick) that adopts the 3-dimensional perspective. (The proof fails in some exceptional configurations, e.g., when A_{1}, A_{2}, B_{1}, and B_{2} are collinear. These are simple enough to be treated individually and will not be considered here. We simply restrict the theorem to two triangles in general position, i.e., assuming that A_{1}≠A_{2}, B_{1}≠B_{2}, C_{1}≠C_{2} and that, similarly, no two corresponding side lines coincide.)

### Proof of Desargues' Theorem

Let O be the common point of A_{1}A_{2}, B_{1}B_{2}, C_{1}C_{2} and suppose , in the first case, that C_{1} is not in the plane OA_{1}B_{1}. Denote the line of intersection of the planes of the triangles by *l*. Since A_{1}A_{2} and B_{1}B_{2} meet at O, they are in the same plane which meets the line *l* in the point ab. In the same way ca and bc must be on *l* as required.

Suppose now that the two triangles are in the same plane. Choose a point E outside the plane and let C_{1}' be a point of the segment between E and C_{1}. The line EC_{2} meets the line OC_{1}' in a point denoted C_{2}'. The proof above then applies to the triangles A_{1}B_{1}C_{1}' and A_{2}B_{2}C_{2}', but seen from E, that is, projecting all the points and lines from E onto the original plane, we get the result for the original triangles.

(Obviously, the proof falls flat lifting C_{1} and C_{2} if A_{1}, B_{1}, A_{2}, B_{2}, are collinear because the new triangles are again coplanar. This can always be avoided by lifting one of the other corresponding pairs.)

### Proof of the converse

Once again, the three-dimensional case is easy. If the triangles are not in the same plane but in planes that intersect in *l*, then, since A_{1}B_{1} and A_{2}B_{2} intersect in ab, they are in the same plane which contains A_{1}A_{2} and B_{1}B_{2}. Similarly, there is a plane containing A_{1}A_{2} and C_{1}C_{2} and one containing B_{1}B_{2} and C_{1}C_{2}. These three planes intersect in the required point O.

In the case when the triangles are in the same plane, we pull them apart as in the previous proof. Choose a point E that is not in the plane and a point A_{1}' between E and A_{1}. Let B_{1}' be the intersection of EB_{1} with abA_{1}' and, similarly, C_{1}' is the intersection of EC_{1} with acA_{1}'. The line B_{1}'C_{1}' will meet *l* in bc as B_{1}C_{1} does. The three-dimensional case then gives the point O' where A_{1}'A_{2}, B_{1}'B_{2} and C_{1}'C_{2} meet and EO' intersects the original plane in the required point O.

### Proof from Menelaus' Theorem

For convenience, the points of intersection of the corresponding sides will be renamed to _{1}C_{1}∩B_{2}C_{2},_{1}A_{1}∩C_{2}A_{2},_{1}B_{1}∩A_{2}B_{2}._{1}C_{1}O, C_{1}A_{1}O, and A_{1}B_{1}O, with the respective transversals B_{2}C_{2}P, C_{2}A_{2}Q, A_{2}B_{2}R, we apply to each pair Menelaus' theorem:

B_{1}P / PC_{1} | · | C_{1}C_{2} / C_{2}O | · | OB_{2} / B_{2}B_{1} | = -1 | |

C_{1}Q / QA_{1} | · | A_{1}A_{2} / A_{2}O | · | OC_{2} / C_{2}C_{1} | = -1 | |

A_{1}R / RB_{1} | · | B_{1}B_{2} / B_{2}O | · | OA_{2} / A_{2}A_{1} | = -1 |

Multiplying the three equalities we obtain after simplification

B_{1}P / PC_{1} | · | C_{1}Q / QA_{1} | · | A_{1}R / RB_{1} | = -1 |

which, by the converse of Menelaus' theorem, implies the collinearity of the three points P, Q, R.

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