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The converse of the theorem implies that a homothety transforms a line in a parallel line. Conversely, the direct statement of the intercept theorem implies that a geometric transformation is always a homothety of center O, if it fixes the lines passing through O and transforms every other line into a parallel line.
The "at most" clause is all that is needed since it can be proved from the remaining axioms that at least one parallel line exists. A proof from ... {3, 3, 3} {3, 3 ...
Thales’ theorem: if AC is a diameter and B is a point on the diameter's circle, the angle ∠ ABC is a right angle.. In geometry, Thales's theorem states that if A, B, and C are distinct points on a circle where the line AC is a diameter, the angle ∠ ABC is a right angle.
To draw the parallel (h) to a diameter g through any given point P. Chose auxiliary point C anywhere on the straight line through B and P outside of BP. (Steiner) In the branch of mathematics known as Euclidean geometry, the Poncelet–Steiner theorem is one of several results concerning compass and straightedge constructions having additional restrictions imposed on the traditional rules.
The proof is written as a series of lines in two columns. In each line, the left-hand column contains a proposition, while the right-hand column contains a brief explanation of how the corresponding proposition in the left-hand column is either an axiom, a hypothesis, or can be logically derived from previous propositions.
The equation of a line in a Euclidean plane is linear, that is, it equates a polynomial of degree one to zero. So, the Bézout bound for two lines is 1, meaning that two lines either intersect at a single point, or do not intersect. In the latter case, the lines are parallel and meet at a point at infinity. One can verify this with equations.
Such an intersection exists if and only if the point P does not belong to the plane (P 1, in green on the figure) that passes through O and is parallel to P 2. It follows that the lines passing through O split in two disjoint subsets: the lines that are not contained in P 1, which are in one to one correspondence with the points of P 2, and ...
The question of the existence of an ordinary line can also be posed for points in the real projective plane RP 2 instead of the Euclidean plane.The projective plane can be formed from the Euclidean plane by adding extra points "at infinity" where lines that are parallel in the Euclidean plane intersect each other, and by adding a single line "at infinity" containing all the added points.