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Since these are equivalent properties, any one of them could be taken as the definition of parallel lines in Euclidean space, but the first and third properties involve measurement, and so, are "more complicated" than the second. Thus, the second property is the one usually chosen as the defining property of parallel lines in Euclidean geometry ...
Because the lines are parallel, the perpendicular distance between them is a constant, so it does not matter which point is chosen to measure the distance. Given the equations of two non-vertical parallel lines
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.
Two pairs of opposite sides are parallel (by definition). Two pairs of opposite sides are equal in length. Two pairs of opposite angles are equal in measure. The diagonals bisect each other. One pair of opposite sides is parallel and equal in length. Adjacent angles are supplementary. Each diagonal divides the quadrilateral into two congruent ...
For a convex quadrilateral with at most two parallel sides, the Newton line is the line that connects the midpoints of the two diagonals. [7] For a hexagon with vertices lying on a conic we have the Pascal line and, in the special case where the conic is a pair of lines, we have the Pappus line. Parallel lines are lines in the same plane that ...
The number of vertices is smaller when some lines are parallel, or when some vertices are crossed by more than two lines. [4] An arrangement can be rotated, if necessary, to avoid axis-parallel lines. After this step, each ray that forms an edge of the arrangement extends either upward or downward from its endpoint; it cannot be horizontal.
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