# Desargues Theorem Proof Using Homogeneous Coordinate

• wawar05
In summary, the conversation discusses the Desargues theorem and a proof using projective geometry. The proof involves starting with four coordinates, D (1,1,1), A (1,0,0), B (0,1,0), and C (0,0,1), and extending lines to form triangles. The question also asks about the coordinates used in the proof and why certain points can be in one line. The expert's response mentions that the approach is similar to the standard proof and suggests using determinants to check for collinearity.
wawar05
Before I ask the question, let me remind that desargues theorem states :

if two triangles are perspective from one point then they are perspective from one line

I'd like to ask whether the order of the steps of the proof I did is correct or not. Since I saw the proof from an article but it only provided the image of the triangles. The proof relates to Projective Geometry

I did the proof of the theorem using homogeneous coordinate which starts with four coordinates D (1,1,1) as the perspective point, A (1,0,0), B (0,1,0), and C (0,0,1). Then ABC forms triangle. Then I extend line DA, DB, and DC and take respectively arbitrary point A' (1,a,a), B' (b,1,b), and C' (c,c,1). Then A'B'C' forms triangle

nb: I just show the beginning steps, since the remains is only algebraic problem and I am not confused on it.

Probably you know why in the article the second triangle has such coordinates A' (1,a,a), B' (b,1,b), and C' (c,c,1)?

additional questions: why (1,1,1), (1,0,0), and (1,a,a) can be in one line?
why (1,1,1), (0,1,0), (b,1,b), can be in one line?
and why (1,1,1), (0,0,1), and (c,c,1) can be in one line?

thank you very much.

Your approach is close to that of the standard proof using projective geometry. You can check for collinearity by determinants.

## 1. What is Desargues Theorem?

Desargues Theorem is a fundamental theorem in projective geometry that states that if two triangles are perspective from a point, then they are perspective from a line. This means that if three pairs of corresponding sides of two triangles intersect at three points, then the three pairs of corresponding sides also intersect at three points when extended to a line.

## 2. What is a proof?

A proof is a logical and mathematical demonstration that shows a statement or theorem to be true. In the context of Desargues Theorem, a proof is a step-by-step explanation of why the theorem holds true in all cases.

## 3. What are homogeneous coordinates?

Homogeneous coordinates are a mathematical tool used in projective geometry to represent points and lines in a coordinate system. They involve adding an extra coordinate to the usual (x,y) Cartesian coordinates, which allows for more flexibility in representing points at infinity and makes it easier to perform geometric transformations and calculations.

## 4. How is Desargues Theorem proved using homogeneous coordinates?

The proof of Desargues Theorem using homogeneous coordinates involves representing the points and lines in the two triangles using homogeneous coordinates, and then using matrix transformations to show that the three points of intersection of corresponding sides also intersect at a common line. The proof also involves using properties of cross ratios and projective transformations.

## 5. Why is Desargues Theorem important?

Desargues Theorem is important because it is a fundamental result in projective geometry that has many applications in mathematics and other fields, such as computer graphics and engineering. It also helps to deepen our understanding of geometric concepts and relationships. Additionally, the proof of Desargues Theorem using homogeneous coordinates is a good exercise in applying matrix transformations and understanding projective geometry concepts.

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