One day two parallel lines fell in love and crossed paths. This is how non-Euclidean geometry came into being! From the diaphragm line to the waist line

CHAPTER 1 LEAVING THE LINE OF POWER (LINE OF ATTACK) This principle is expressed folk wisdom: “Don’t get into trouble.” Rozhon is a stake that a fool goes for directly, that is, head-on. In general, in life, a frontal attack, literally and figuratively, is a thankless and very traumatic task. At

This article is about parallel lines and parallel lines. First, the definition of parallel lines on a plane and in space is given, notations are introduced, examples and graphic illustrations of parallel lines are given. Next, the signs and conditions for parallelism of lines are discussed. The conclusion shows solutions characteristic tasks to prove the parallelism of lines that are given by some line equations in rectangular system coordinates on the plane and in three-dimensional space.

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Parallel lines - basic information.

Definition.

Two lines in a plane are called parallel if they don't have common points.

Definition.

Two lines in three-dimensional space are called parallel, if they lie in the same plane and do not have common points.

Please note that the clause “if they lie in the same plane” in the definition of parallel lines in space is very important. Let us clarify this point: two lines in three-dimensional space that do not have common points and do not lie in the same plane are not parallel, but intersecting.

Here are some examples of parallel lines. The opposite edges of the notebook sheet lie on parallel lines. The straight lines along which the plane of the wall of the house intersects the planes of the ceiling and floor are parallel. Railroad rails on level ground can also be considered as parallel lines.

To denote parallel lines, use the symbol “”. That is, if lines a and b are parallel, then we can briefly write a b.

Please note: if lines a and b are parallel, then we can say that line a is parallel to line b, and also that line b is parallel to line a.

Let's voice the statement that plays important role when studying parallel lines on a plane: through a point not lying on a given line, there passes a single straight line parallel to the given one. This statement is accepted as a fact (it cannot be proven on the basis of the known axioms of planimetry), and it is called the axiom of parallel lines.

For the case in space, the theorem is valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem is easily proven using the above axiom of parallel lines (you can find its proof in the geometry textbook for grades 10-11, which is listed at the end of the article in the list of references).

For the case in space, the theorem is valid: through any point in space that does not lie on a given line, there passes a single straight line parallel to the given one. This theorem can be easily proven using the above parallel line axiom.

Parallelism of lines - signs and conditions of parallelism.

A sign of parallelism of lines is sufficient condition parallelism of lines, that is, a condition the fulfillment of which guarantees parallelism of lines. In other words, the fulfillment of this condition is sufficient to establish the fact that the lines are parallel.

There are also necessary and sufficient conditions for the parallelism of lines on a plane and in three-dimensional space.

Let us explain the meaning of the phrase “necessary and sufficient condition for parallel lines.”

We have already dealt with the sufficient condition for parallel lines. And what is “ necessary condition parallelism of lines"? From the name “necessary” it is clear that the fulfillment of this condition is necessary for parallel lines. In other words, if the necessary condition for the lines to be parallel is not met, then the lines are not parallel. Thus, necessary and sufficient condition for parallel lines is a condition the fulfillment of which is both necessary and sufficient for parallel lines. That is, on the one hand, this is a sign of parallelism of lines, and on the other hand, this is a property that parallel lines have.

Before formulating a necessary and sufficient condition for the parallelism of lines, it is advisable to recall several auxiliary definitions.

Secant line is a line that intersects each of two given non-coinciding lines.

When two straight lines intersect with a transversal, eight undeveloped ones are formed. In the formulation of the necessary and sufficient condition for the parallelism of lines, the so-called lying crosswise, corresponding And one-sided angles. Let's show them in the drawing.

Theorem.

If two straight lines in a plane are intersected by a transversal, then for them to be parallel it is necessary and sufficient that the intersecting angles be equal, or corresponding angles were equal, or the sum of one-sided angles was equal to 180 degrees.

Let us show a graphic illustration of this necessary and sufficient condition for the parallelism of lines on a plane.

You can find proofs of these conditions for the parallelism of lines in geometry textbooks for grades 7-9.

Note that these conditions can also be used in three-dimensional space - the main thing is that the two straight lines and the secant lie in the same plane.

Here are a few more theorems that are often used to prove the parallelism of lines.

Theorem.

If two lines in a plane are parallel to a third line, then they are parallel. The proof of this criterion follows from the axiom of parallel lines.

There is a similar condition for parallel lines in three-dimensional space.

Theorem.

If two lines in space are parallel to a third line, then they are parallel. The proof of this criterion is discussed in geometry lessons in the 10th grade.

Let us illustrate the stated theorems.

Let us present another theorem that allows us to prove the parallelism of lines on a plane.

Theorem.

If two lines in a plane are perpendicular to a third line, then they are parallel.

There is a similar theorem for lines in space.

Theorem.

If two lines in three-dimensional space are perpendicular to the same plane, then they are parallel.

Let us draw pictures corresponding to these theorems.

All the theorems, criteria and necessary and sufficient conditions formulated above are excellent for proving the parallelism of lines using geometry methods. That is, to prove the parallelism of two given lines, you need to show that they are parallel to a third line, or show the equality of crosswise lying angles, etc. A bunch of similar tasks solved in geometry lessons in high school. However, it should be noted that in many cases it is convenient to use the coordinate method to prove the parallelism of lines on a plane or in three-dimensional space. Let us formulate the necessary and sufficient conditions for the parallelism of lines that are specified in a rectangular coordinate system.

Parallelism of lines in a rectangular coordinate system.

In this paragraph of the article we will formulate necessary and sufficient conditions for parallel lines in a rectangular coordinate system, depending on the type of equations defining these straight lines, and we also present detailed solutions characteristic tasks.

Let's start with the condition of parallelism of two straight lines on a plane in the rectangular coordinate system Oxy. His proof is based on the definition of the direction vector of a line and the definition of the normal vector of a line on a plane.

Theorem.

For two non-coinciding lines to be parallel in a plane, it is necessary and sufficient that the direction vectors of these lines are collinear, or the normal vectors of these lines are collinear, or the direction vector of one line is perpendicular to the normal vector of the second line.

Obviously, the condition of parallelism of two lines on a plane is reduced to (direction vectors of lines or normal vectors of lines) or to (direction vector of one line and normal vector of the second line). Thus, if and are direction vectors of lines a and b, and And are normal vectors of lines a and b, respectively, then the necessary and sufficient condition for the parallelism of lines a and b will be written as , or , or , where t is some real number. In turn, the coordinates of the guides and (or) normal vectors of straight lines a and b are found by known equations straight

In particular, if straight line a in the rectangular coordinate system Oxy on the plane defines a general straight line equation of the form , and straight line b - , then the normal vectors of these lines have coordinates and, respectively, and the condition for parallelism of lines a and b will be written as .

If line a corresponds to the equation of a line with an angular coefficient of the form , and line b - , then the normal vectors of these lines have coordinates and , and the condition for parallelism of these lines takes the form . Consequently, if straight lines on a plane in a rectangular coordinate system are parallel and can be specified by equations of straight lines with angular coefficients, then slopes straight lines will be equal. And conversely: if non-coinciding lines on a plane in a rectangular coordinate system can be specified by the equations of a line with equal angular coefficients, then such lines are parallel.

If a line a and a line b in a rectangular coordinate system are determined by the canonical equations of a line on a plane of the form And , or parametric equations of a straight line on a plane of the form And accordingly, the direction vectors of these lines have coordinates and , and the condition for parallelism of lines a and b is written as .

Let's look at solutions to several examples.

Example.

Are the lines parallel? And ?

Solution.

Let us rewrite the equation of a straight line in segments in the form general equation straight: . Now we can see that is the normal vector of the line , a is the normal vector of the line. These vectors are not collinear, since there is no such real number t for which the equality ( ). Consequently, the necessary and sufficient condition for the parallelism of lines on a plane is not satisfied, therefore, the given lines are not parallel.

Answer:

No, the lines are not parallel.

Example.

Are straight lines and parallel?

Solution.

Let's give canonical equation straight line to the equation of a straight line with an angular coefficient: . Obviously, the equations of the lines and are not the same (in this case, the given lines would be the same) and the angular coefficients of the lines are equal, therefore, the original lines are parallel.

attention and thus, as it were, sealing them, are known to seers as parallel lines .

Passing through parallel lines

Don Juan explained to me that the study of the second attention must begin with the awareness that the force of the First Ring of Power, which limits us, is a physical, concrete barrier. Seers have described it as a wall of fog, a barrier that can be systematically perceived as a result of the First Ring of Power being blocked; and which can then be overcome through warrior training.

When warriors penetrate the wall of fog, they feel their bodies being squeezed, or they feel intense shaking inside their bodies, mainly to the right of the abdomen or through the midsection, from right to left. .

There is an area between both lines special condition consciousness, which seers call limbo, or the world between parallel lines. This is the real space between two huge layers of Eagle emanations; emanations that lie within human capabilities awareness. One of these layers of emanations is the layer that creates the world Everyday life, and the second layer creates other worlds of second attention. So limbo is transition zone, in which both fields of emanations extend to each other.

The feeling of heaviness, Don Juan continued, felt in limbo arises from the increasing pressure exerted on the First Attention. In the area directly behind the wall of fog, we can still behave as normal; as if we are in a grotesque but recognizable world. As we penetrate further beyond the wall of fog, with each step it becomes more difficult to discern details or behave in a normal way.

From the moment the First Ring of Power begins to create scanning objects, the way they are formed becomes our natural perceptual predisposition. Interrupting the unifying force of the First Attention means crossing the first boundary line. The normal perceptual predisposition falls in the intermediate region between the parallel lines. At the same time, we continue to build almost normal scan images for some time. But as soon as we approach what seers call the second boundary line, the perceptual predisposition of the First Attention begins to recede, to lose strength. Don Juan said that such a transition is marked by a sudden inability to remember or understand what is happening. As one approaches the second boundary line, the second attention begins to influence the warriors who have undertaken such a journey more and more strongly. If they are inexperienced, their awareness becomes empty, empty. Don Juan believed that this was happening because they were approaching the spectrum of the Eagle's emanations, which did not yet have a systematized perceptual predisposition for them. My experiences with La Gorda and the Nagual woman behind the wall of fog are an example of this inability. I traveled far to the other side, but I could not explain what we did there, for the simple reason that my second attention had not yet formed, and this did not give me the opportunity to formulate everything that I perceived there.

Don Juan explained to me that we are beginning to activate the Second Ring of Power, forcing the second attention to awaken from sleep. This is achieved by blocking the action of the First Ring of Power. Then, the teacher's task is to recreate the state from which the First Ring of Power started, a state saturated with intention. The first Ring of Power was launched by the power of intention given by those who taught us the art of scanning. As my teacher, he gave me a new intention, which was to create a new sphere of perception.

As the warriors penetrate the second line, they feel a sharp cracking sound in their upper body, something like the sound of a dry branch breaking.

Don Juan said what is required whole life continuous discipline, which seers call unbending intention, to prepare the Second Ring of Power to be able to create scan images corresponding to another level of Eagle emanations. To master the predisposition of perception of both parallel worlds is a feat of incomparable significance, which only a few warriors achieve. Silvio Manuel was one of those few.

Don Juan warned me not to try to master this deliberately. If this happens, it must happen through a natural process that unfolds itself without special effort from our side. He explained to me that the reason for such impartiality was based on the practical considerations that when it is achieved forcefully, it later turns out to be difficult to give it up, while the goal to which warriors should strive should be to overcome both predispositions of perception, to enter the final freedom of the third attention.

Connections

Connection with loss of human form

Somewhere nearby, in the mountains. The Nagual said that there is a natural crack at this place. He said that certain places of power are holes in this world. If you are formless, you can go through such a hole into the unknown, into another world. That world and this one - where we live - are on two parallel lines. It is quite possible that he different time took me into that world through these lines, but we don’t remember this.

And with the second attention there are two dividing boundaries, between which there is a special transition region attention, called Limbo.

They extend into infinity, never allowing themselves to be bypassed and can only be crossed. The crossing of parallel lines is identical to overcoming this particular barrier of perception.

It is also important to understand that Parallel Lines are essentially the positions of the assemblage point. More precisely, the intersection of each of the lines are the positions of the assemblage point.

The nature of parallel lines

The position of the assemblage point at which a dense formation is formed energy body(or in other words the dream body, the luminous body), the so-called Double or Double, is very close to the position of intersection of parallel lines.

An obstacle to its formation, and therefore to the perception of other worlds (equivalent to getting there), is the barrier of perception (see the article Barrier of Perception for more details), which represents in this case manifestation of the power of holding perception with the First Ring of Power in the first attention mode.

This barrier consists of two special boundaries parallel to each other, crossing the first of which usually leads to entering the world between the lines (Limbo), and crossing the second leads to entering one of the worlds of the second attention.

It is also possible to pass both borders at once and directly enter another world, but this requires much more from the warrior. greater skill and experience.

Perception of parallel lines

If the first line, due to the lateral shift of the assemblage point, can be perceived both visually (in the form of a wall of yellow fog) and at the bodily level, then the perception of the second line is available only at the “bodily” level.

When crossing the first dividing line you feel:

1. The body is compressed, compressed

2. The body experiences severe tremors and shakes.

3. The epicenter of sensations is mainly to the right of the abdomen or, through the middle part, from right to left.

When crossing the second dividing line you feel:

1. A sharp cracking sound in the upper part of the body (something like the sound of a breaking branch).

2. Passing through a tight paper screen.

2 On your own.

Training, naturally, begins with the first method so that the adept can subsequently use the accumulated knowledge and be able to cross parallel lines himself. Moreover, the student, due to the fact that the crossing of lines occurs in a very strongly altered state of consciousness, does not even remember how he was forced to cross the lines and only remembers everything as he learns.

Even the poles are connected by meridians.
What can we say about parallels,
which no, no, and will intersect...

If people meet, going towards each other, it means they have different paths...
.Axiom..
Beloved Woman is a daily theorem of love and the only axiom of male happiness... A woman is sure that if she likes a man, then their parallel paths must intersect... No one will argue...
Mathematics taught us long ago that two parallel lines never intersect. Mathematics is completely indifferent to this, but people sometimes incorrectly interpret all the laws, including mathematical ones, trying them on in their lives...
The two destinies exist independently of each other. Living with their joys and sorrows until they intersect. Perhaps they had seen each other before, known each other by sight, heard each other’s voice.
Perhaps they even lived on the same floor or in the same city. But they didn’t matter to each other, until one day their straight lines intersected. In the very literally! They collided at the entrance, stepped on each other's feet, ended up in the same line, met at a party...
Whatever you like, a point of contact has appeared. Trajectories have changed. They nudged each other, froze for a second, and... liked each other. What happens to the straight lines? The movement cannot stop; if it stops, everything will end. Only now they will try to move together, in the same direction....
Direction! That's the most important thing! If one line stretched from left to right, and the other from top to bottom, how would they connect? No way. They will stay together for some time, at one point, and soon, each of them will continue to move along its own trajectory.... If people are looking for happiness in its various manifestations, if She dreams of becoming a dancer, and He dreams of flying into space. If He deals with finances and she is a housewife. If she hates yellow, and He wears only these shades, this does not mean that they will not succeed. This means that they are a little different. It is important that they always look in the same direction. To the future, or to the sky, or to the sunset... Let there be one goal, but the ways to achieve it may vary. This will not change the meaning, content, but form is a relative concept....
And also... one line should not overlap another. They can stretch parallel, but very close, touching the edges. Stretching like this into infinity... Yes, it happens... I know... Two Parallel ones intersect in infinity - and they themselves believe in it.
The main thing is to meet....by the will of Lady Chance... It doesn’t matter where or how....
Don't pass by....
P.S. IMHO... but sometimes two parallel ones intersecting form a cross... a cross is placed on everything... for some it forms a cross, and for others a dot... and then the parallel ones go nowhere... and so it happens.. This happens most often...for many...

Parallel radiant glow.. Pair family lines.. their passion is so strong... And as the fruit of that intersection... A small dot was born!......

Parallel lines do not intersect.. The axiom sounds doomed.. They will never. Never... Meet.. Parallel engaged.. Betrothed, betrothed, parallel.. Reaching into the beyond.. Parallel lines as a rule! Not in time, and not in this limb.. They will not come together in cheerful carelessness.. No matter how close life puts them... And how close they are not drawn.. There are no points for them to intersect... Arguing with the rules is risky.. Here such a statement! Whoever doesn’t understand, doesn’t need it... And whoever understands, my brother is in misfortune... There is no cure for lost love - Better than friendly participation! Better new love, unexpected.. Hot glances, affectionate hugs.. Options are given to us from above.. You will not escape significant events... I wish everyone a blue sky.. Happiness, joy and luck.. And to the lines broken by life.. More points of intersection! Well, we will forever remain.. The inaccessibility of renunciation... Melting in the yellow flame of a candle.. Only our trace from the intersection....