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One approach to cross ratio interprets it as a homography that takes three designated points to 0, 1, and ∞. Under restrictions having to do with inverses, it is possible to generate such a mapping with ring operations in the projective line over a ring. The cross ratio of four points is the evaluation of this homography at the fourth point.
Pappus theorem: proof If the affine form of the statement can be proven, then the projective form of Pappus's theorem is proven, as the extension of a pappian plane to a projective plane is unique. Because of the parallelity in an affine plane one has to distinct two cases: g ∦ h {\displaystyle g\not \parallel h} and g ∥ h {\displaystyle g ...
A short elementary proof of Pascal's theorem in the case of a circle was found by van Yzeren (1993), based on the proof in (Guggenheimer 1967). This proof proves the theorem for circle and then generalizes it to conics. A short elementary computational proof in the case of the real projective plane was found by Stefanovic (2010).
Define the cross-ratio ... Proof: It is known that the area of a triangle inscribed in a circle of radius is: = Writing the area of the quadrilateral as sum of two ...
The cross-ratio (,;,) = () is a ratio of division ratios, or a double ratio. Setting the double ratio to minus one means that when t(c) + t(d) = 0, then c and d are harmonic conjugates with respect to a and b. So the division ratio criterion is that they be additive inverses.
In the same way we may find a definition of the product of two throws. As the product of two numbers bears the same ratio to one of them as the other bears to unity, the ratio of two numbers is the cross ratio which they as a pair bear to infinity and zero, so Von Staudt, in the previous notation, defines the product of two throws by
A proof given by John Wellesley Russell uses Pasch's axiom to consider cases where a line does or does not meet a triangle. [4] First, the sign of the left-hand side will be negative since either all three of the ratios are negative, the case where the line DEF misses the triangle (see diagram), or one is negative and the other two are positive, the case where DEF crosses two sides of the ...
Two points z 1 and z 2 are conjugate with respect to a generalized circle C, if, given a generalized circle D passing through z 1 and z 2 and cutting C in two points a and b, (z 1, z 2; a, b) are in harmonic cross-ratio (i.e. their cross ratio is −1). This property does not depend on the choice of the circle D.