The proof is by construction: an equilateral triangle ΑΒΓ is made by drawing circles Δ and Ε centered on the points Α and Β, and taking one intersection of the circles as the third vertex of the triangle. Postulates 3 and 5 hold only for plane geometry in three dimensions, postulate 3 defines a sphere.Ī proof from Euclid's elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides. The circle described in postulate 3 is tacitly unique. The following verbs appear: join, extend, draw, intersect. These axioms invoke the following concepts: point, straight line segment and line, side of a line, circle with radius and centre, right angle, congruence, inner and right angles, sum. If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
0 Comments
Leave a Reply. |