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If two lines (a and b) are both perpendicular to a third line (c), all of the angles formed along the third line are right angles. Therefore, in Euclidean geometry , any two lines that are both perpendicular to a third line are parallel to each other, because of the parallel postulate .
The line with equation ax + by + c = 0 has slope -a/b, so any line perpendicular to it will have slope b/a (the negative reciprocal). Let ( m , n ) be the point of intersection of the line ax + by + c = 0 and the line perpendicular to it which passes through the point ( x 0 , y 0 ).
Here, p is the (positive) length of the line segment perpendicular to the line and delimited by the origin and the line, and is the (oriented) angle from the x-axis to this segment. It may be useful to express the equation in terms of the angle α = φ + π / 2 {\displaystyle \alpha =\varphi +\pi /2} between the x -axis and the line.
In three or more dimensions, even two lines almost certainly do not intersect; pairs of non-parallel lines that do not intersect are called skew lines. But if an intersection does exist it can be found, as follows. In three dimensions a line is represented by the intersection of two planes, each of which has an equation of the form
Carnot's theorem: if three perpendiculars on triangle sides intersect in a common point F, then blue area = red area. Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection.
The straight lines which form right angles are called perpendicular. [8] Euclid uses right angles in definitions 11 and 12 to define acute angles (those smaller than a right angle) and obtuse angles (those greater than a right angle). [9] Two angles are called complementary if their sum is a right angle. [10] Book 1 Postulate 4 states that all ...
The line k is a variable line through K. The line l through L is perpendicular to k. The angle between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.
In elliptic geometry, two lines perpendicular to a given line must intersect. In fact, all perpendiculars to a given line intersect at a single point called the absolute pole of that line. Every point corresponds to an absolute polar line of which it is the absolute pole. Any point on this polar line forms an absolute conjugate pair with the