The following remarkable theorem is named after Gerard Desargues (1591-1661) who was a French architect and geometer, and discovered it in 1639.
Desargues' Theorem
Two triangles ABC and A'B'C' are point perspective (in O), if and only if, they are line perspective (in line XYZ).
Desargues' Theorem
Challenge
1) Can you prove Desargues' theorem? First try and solve it yourself before looking at one possible solution given below
Solution: Surprisingly, the proof of the theorem is simplified by considering the 3D case, e.g. click on the 'Link to Desargues 3D' button to navigate to a new sketch showing the theorem in 3D. Then read the short proof given in the paragraph Solving 2D by going 3D in De Villiers & Garner (2008) - see reference (and link below). For a more detailed proof and discussion, also see Lange (2015) - link below.
Some Applications
2) Desargues' theorem provides a useful method for perspective image construction, and was a consequence of the development of the powerful idea of perspective drawing in paintings from the Renaissance period onwards. It also paved the way, together with the much older theorem of Pappus, and the theorem of Pascal, to the development of projective geometry in the 19th century by mathematicians like Poncelet, Monge and others. For more about perspective or descriptive geometry, read the excellent free book by Mladinić & Radović (2019) - see reference (and link below).
3) With Desargues' theorem, one can easily (almost trivially) prove the concurrency of the medians as shown at Concurrency of medians with Desargues.
4) The theorem is also effectively used in the proof of the following Interesting Collinearity.
5) Here is a valuable PDF compilation of problems, solutions and papers by Chris Bradley, which contain several applications of Desargues: 259 Geometry Writings by Christopher Bradley.
6) Desargues' Theorem also beautifully displays the projective duality between "points" and "lines", and between "collinearity" and "concurrency".
a) Do a Google search to read up more on "projective duality".
b) For a poem of mine written many years ago related to the topic of duality and specifically referring to Desargues, go to Duality (a Poem).
References
De Villiers, M. & Garner, M. (2008). Problem solving and proving via generalisation. Learning and Teaching Mathematics, No. 5 (May), pp. 19-25.
Lange, M. (2015). Explanation, Existence and Natural Properties in Mathematics – A Case Study: Desargues’ Theorem. dialectica, Vol. 69, N° 4, pp. 435–472.
Mladinić, P. & Radović, N. (2019). Descriptive Geometry (Perspective, Monge's Procedure, Axonometry). Publisher: PROVEN Grupa d.o.o., Zagreb, Croatia.
Related Links
Concurrency Conjecture (Rethinking Proof activity)
The Center of Gravity of a Triangle (Rethinking Proof activity - concurrency of medians, Ceva's theorem)
Proof of Concurrency of medians with Desargues
An Interesting Collinearity
The Fermat-Torricelli Point (Rethinking Proof activity)
Airport Problem (Rethinking Proof activity)
Triangle Altitudes (Rethinking Proof activity)
Experimentally Finding the Medians and Centroid of a Triangle
Experimentally Finding the Centroid of a Triangle with Different Weights at the Vertices (Ceva's theorem)
A side trisection triangle concurrency
A 1999 British Mathematics Olympiad Problem and its dual
Anghel's Hexagon Concurrency theorem
External Links
Desargues's theorem (Wikipedia)
Monge from Desargues (Cut The Knot)
Desargues' Theorem (Cut The Knot)
Perspective (geometry) (Wikipedia)
Desargues’s theorem (Brittanica)
Desargues’s theorem (Wolfram MathWorld)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
SA Mathematics Olympiad Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)
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Michael de Villiers, created with JavaSketchpad in Oct 2005; updated 31 October 2010; converted to WebSketchpad, 9 Oct 2025; updated 29 Sept 2026.