What is kilian triangle?

The Kilian triangle is a geometric construction in projective geometry named after the German mathematician Kilian. It consists of three lines that intersect at a point, with each line intersecting the opposite side of a triangle at a point.

The Kilian triangle has several properties that make it useful in projective geometry. One such property is that any line that intersects two sides of a triangle also intersects the third side at a point that lies on the Kilian triangle. This property can be used to prove various theorems in projective geometry, such as Desargues' theorem and Pappus's theorem.

The Kilian triangle is also related to the concept of harmonic conjugates. If a line intersects two sides of a triangle at points A and B, then the Kilian point of that line is the point at which the third side intersects the Kilian triangle. The four points A, B, the Kilian point, and the point at infinity are harmonic conjugates. This fact can be used to prove the cross-ratio formula in projective geometry.

Overall, the Kilian triangle is an important tool in projective geometry that provides a useful framework for studying the properties of triangles and lines in a projective space.