© Kugusheva Natalya Lvovna, 2009 Geometry, 8th grade TRIANGLE FOUR REMARKABLE POINTS
The intersection point of the medians of a triangle The intersection point of the bisectors of a triangle The intersection point of the altitudes of a triangle The intersection point of the perpendicular bisectors of a triangle
The median (BD) of a triangle is the segment that connects the vertex of the triangle to the midpoint of the opposite side. A B C D Median
The medians of a triangle intersect at one point (the center of gravity of the triangle) and are divided by this point in a ratio of 2: 1, counting from the vertex. AM: MA 1 = VM: MV 1 = SM:MS 1 = 2:1. A A 1 B B 1 M C C 1
The bisector (A D) of a triangle is the bisector segment of the interior angle of the triangle.
Each point of the bisector of an undeveloped angle is equidistant from its sides. Conversely: every point lying inside an angle and equidistant from the sides of the angle lies on its bisector. A M B C
All bisectors of a triangle intersect at one point - the center of the circle inscribed in the triangle. C B 1 M A V A 1 C 1 O The radius of a circle (OM) is a perpendicular dropped from the center (TO) to the side of the triangle
HEIGHT The altitude (C D) of a triangle is the perpendicular segment drawn from the vertex of the triangle to the straight line containing the opposite side. A B C D
The altitudes of a triangle (or their extensions) intersect at one point. A A 1 B B 1 C C 1
MIDPERPENDICULAR The perpendicular bisector (DF) is the line perpendicular to the side of the triangle and dividing it in half. A D F B C
A M B m O Each point of the perpendicular bisector (m) to a segment is equidistant from the ends of this segment. Conversely: every point equidistant from the ends of a segment lies on the perpendicular bisector to it.
All perpendicular bisectors of the sides of the triangle intersect at one point - the center of the circle circumscribed about the triangle. A B C O The radius of the circumscribed circle is the distance from the center of the circle to any vertex of the triangle (OA). m n p
Tasks for students Construct a circle inscribed in an obtuse triangle using a compass and ruler. To do this: Construct bisectors in an obtuse triangle using a compass and ruler. The point of intersection of the bisectors is the center of the circle. Construct the radius of the circle: a perpendicular from the center of the circle to the side of the triangle. Construct a circle inscribed in the triangle.
2. Using a compass and ruler, construct a circle circumscribing an obtuse triangle. To do this: Construct perpendicular bisectors to the sides of the obtuse triangle. The point of intersection of these perpendiculars is the center of the circumscribed circle. The radius of a circle is the distance from the center to any vertex of the triangle. Construct a circle around the triangle.
The 8th grade geometry lesson is designed based on the positional learning model.
Lesson objectives:
- Studying theoretical material on the topic “Four remarkable points of a triangle”;
- Development of thinking, logic, speech, imagination of students, ability to analyze and evaluate work;
- Development of group work skills;
- Fostering a sense of responsibility for the quality and results of the work performed.
Equipment:
- cards with group names;
- cards with tasks for each group;
- A-4 paper for recording the results of the groups’ work;
- epigraph written on the board.
During the classes
1. Organizational moment.
2. Determining the goals and topic of the lesson.
Historically, geometry began with a triangle, so for two and a half millennia the triangle has been a symbol of geometry. School geometry can only become interesting and meaningful, only then can it become geometry proper when it includes a deep and comprehensive study of the triangle. Surprisingly, the triangle, despite its apparent simplicity, is an inexhaustible object of study - no one, even in our time, dares to say that they have studied and know all the properties of the triangle.
Who hasn't heard about the Bermuda Triangle, in which ships and planes disappear without a trace? But the triangle itself is fraught with a lot of interesting and mysterious things.
The central place of the triangle is occupied by the so-called remarkable points.
I think that at the end of the lesson you will be able to say why the points are called remarkable and whether they are so.
What is the topic of our lesson? "Four remarkable points of the triangle." The epigraph for the lesson can be the words of K. Weierstrass: “A mathematician who is not partly a poet will never achieve perfection in mathematics” (the epigraph is written on the board).
Look at the wording of the topic of the lesson, at the epigraph and try to determine the goals of your work in the lesson. At the end of the lesson we will check how well you have completed them.
3. Independent work of students.
Preparing for independent work
To work in the lesson, you must choose one of six groups: “Theorists”, “Creativity”, “Logic-designers”, “Practitioners”, “Historians”, “Experts”.
Briefing
Each group receives task cards. If the task is not clear, the teacher provides additional explanations.
"Theorists"
Assignment: define the basic concepts necessary when studying the topic “Four remarkable points of a triangle” (altitude of a triangle, median of a triangle, bisector of a triangle, perpendicular bisector, incircle, circumcircle), you can use a textbook; write the main concepts on a piece of paper.
"Historians"
bisectors center of the inscribed circle perpendiculars the center of the circumscribed circle. The Principia does not say that even three heights triangles intersect at one point called orthocenter median center of gravity
In the 20s of the XIX century. French mathematicians J. Poncelet, C. Brianchon and others independently established the following theorem: the bases of medians, the bases of altitudes and the midpoints of segments of altitudes connecting the orthocenter with the vertices of the triangle lie on the same circle.
This circle is called the “circle of nine points”, or “Feuerbach circle”, or “Euler circle”. K. Feuerbach established that the center of this circle lies on the “Euler straight line”.
Assignment: analyze the article and fill out a table reflecting the material studied.
Point name |
What intersects |
||
"Creation"
Assignment: come up with syncwine(s) on the topic “Four remarkable points of a triangle” (for example, triangle, point, median, etc.)
Rule for writing syncwine:
In the first line, the topic is named in one word (usually a noun).
The second line is a description of the topic in two words (2 adjectives).
The third line is a description of the action within the framework of this topic in three words (verbs, gerunds).
The fourth line is a 4-word phrase showing the attitude towards the topic.
The last line is a one-word synonym (metaphor) that repeats the essence of the topic.
"Logic constructors"
The median of a triangle is a segment connecting any vertex of the triangle with the midpoint of the opposite side. Any triangle has three medians.
A bisector is a segment of the bisector of any angle from the vertex to the intersection with the opposite side. Any triangle has three bisectors.
The altitude of a triangle is the perpendicular drawn from any vertex of the triangle to the opposite side or its extension. Any triangle has three heights.
The perpendicular bisector of a segment is a line passing through the middle of a given segment and perpendicular to it. Any triangle has three perpendicular bisectors.
Assignment: Using triangular sheets of paper, construct by folding the intersection points of medians, altitudes, bisectors, and bisectors. Explain this to the whole class.
"Practices"
In the fourth book of Elements, Euclid solves the problem “Inscribing a circle in a given triangle.” From the solution it follows that three bisectors The interior angles of a triangle intersect at one point - center of the inscribed circle. From the solution of another Euclidean problem it follows that perpendiculars, restored to the sides of the triangle at their midpoints, also intersect at one point - the center of the circumscribed circle. The Principia does not say that the three altitudes of a triangle intersect at one point called orthocenter(The Greek word “orthos” means straight, correct). This proposal, however, was known to Archimedes, Pappus, and Proclus. The fourth singular point of a triangle is the intersection point median. Archimedes proved that she is center of gravity(barycenter) of the triangle. Particular attention was paid to the above four points starting from the 18th century. They were called "remarkable" or "special points of the triangle."
The study of the properties of a triangle associated with these and other points served as the beginning for the creation of a new branch of elementary mathematics - “triangle geometry”, or “new triangle geometry”, one of the founders of which was Leonhard Euler.
Assignment: analyze the proposed material and come up with a diagram that reflects the semantic connections between the units, explain it, draw it on a piece of paper, and display it on the board.
Remarkable points of the triangle
1.____________ 2.___________ 3.______________ 4.____________
Drawing 1 Drawing 2 Drawing 3 Drawing 4
____________ ___________ ______________ ____________
(explanation)
"Experts"
Assignment: make a table in which you evaluate the work of each group, select the parameters by which you will evaluate the work of the groups, determine the points.
The parameters can be the following: participation of each student in the work of his group, participation in the defense, interesting presentation of the material, presentation of clarity, etc.
In your speech, you should note the positive and negative aspects in the activities of each group.
4. Group performance.(2-3 minutes each)
The results of the work are posted on the board
5. Summing up the lesson.
Look at the goals you set at the beginning of the lesson. Did you manage to complete everything?
Do you agree with the epigraph that was chosen for today's lesson?
6. Homework assignment.
1) Make sure that the triangle, which rests on the tip of the needle at a certain point, is in balance, using the material from today's lesson.
2) Draw all 4 remarkable points in different triangles.
FOUR NOTABLE POINTS
TRIANGLE
Geometry
8th grade
Sakharova Natalia Ivanovna
MBOU Secondary School No. 28 of Simferopol
- Intersection point of triangle medians
- Intersection point of triangle bisectors
- Point of intersection of triangle altitudes
- Point of intersection of the perpendicular medians of a triangle
Median
Median (BD) of a triangle is the segment that connects the vertex of the triangle to the midpoint of the opposite side.
Medians triangles intersect at one point (center of gravity triangle) and are divided by this point in a ratio of 2: 1, counting from the vertex.
BISECTOR
Bisector (AD) of a triangle is the bisector segment of the interior angle of the triangle. ∟ BAD = ∟CAD.
Every point bisectors of an undeveloped angle is equidistant from its sides.
Back: every point lying inside an angle and equidistant from the sides of the angle lies on its bisector.
All bisectors triangles intersect at one point - center of the inscribed into a triangle circles.
The radius of the circle (OM) is a perpendicular descended from the center (TO) to the side of the triangle
HEIGHT
Height (CD) of a triangle is a perpendicular segment drawn from a vertex of the triangle onto a line containing the opposite side.
Heights triangles (or their extensions) intersect one point.
MIDDLE PERPENDICULAR
Perpendicular bisector (DF) called a straight line perpendicular to a side of a triangle and dividing it in half.
Every point perpendicular bisector(m) to a segment is equidistant from the ends of this segment.
Back: every point equidistant from the ends of a segment lies on the midpoint perpendicular to him.
All perpendicular bisectors of the sides of a triangle intersect at one point - the center of the described near the triangle circle .
The radius of the circumcircle is the distance from the center of the circle to any vertex of the triangle (OA).
Page 177 No. 675 (drawing)
Homework
P. 173 § 3 definitions and theorems p. 177 No. 675 (finish)
In this lesson we will look at four wonderful points of the triangle. Let us dwell on two of them in detail, recall the proofs of important theorems and solve the problem. Let us remember and characterize the remaining two.
Subject:Revision of the 8th grade geometry course
Lesson: Four Wonderful Points of a Triangle
A triangle is, first of all, three segments and three angles, therefore the properties of segments and angles are fundamental.
The segment AB is given. Any segment has a midpoint, and a perpendicular can be drawn through it - let’s denote it as p. Thus, p is the perpendicular bisector.
Theorem (main property of the perpendicular bisector)
Any point lying on the perpendicular bisector is equidistant from the ends of the segment.
Prove that
Proof:
Consider triangles and (see Fig. 1). They are rectangular and equal, because. have a common leg OM, and legs AO and OB are equal by condition, thus, we have two right triangles, equal in two legs. It follows that the hypotenuses of the triangles are also equal, that is, what was required to be proved.
Rice. 1
The converse theorem is true.
Theorem
Each point equidistant from the ends of a segment lies on the perpendicular bisector to this segment.
Given a segment AB, a perpendicular bisector to it p, a point M equidistant from the ends of the segment (see Fig. 2).
Prove that point M lies on the perpendicular bisector of the segment.
Rice. 2
Proof:
Consider a triangle. It is isosceles, as per the condition. Consider the median of a triangle: point O is the middle of the base AB, OM is the median. According to the property of an isosceles triangle, the median drawn to its base is both an altitude and a bisector. It follows that . But line p is also perpendicular to AB. We know that at point O it is possible to draw a single perpendicular to the segment AB, which means that the lines OM and p coincide, it follows that the point M belongs to the straight line p, which is what we needed to prove.
If it is necessary to describe a circle around one segment, this can be done, and there are infinitely many such circles, but the center of each of them will lie on the perpendicular bisector to the segment.
They say that the perpendicular bisector is the locus of points equidistant from the ends of a segment.
A triangle consists of three segments. Let us draw bisectoral perpendiculars to two of them and obtain the point O of their intersection (see Fig. 3).
Point O belongs to the perpendicular bisector to side BC of the triangle, which means it is equidistant from its vertices B and C, let’s denote this distance as R: .
In addition, point O is located on the perpendicular bisector to segment AB, i.e. , at the same time, from here.
Thus, point O of the intersection of two midpoints
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perpendiculars of the triangle is equidistant from its vertices, which means it also lies on the third bisector perpendicular.
We have repeated the proof of an important theorem.
The three perpendicular bisectors of a triangle intersect at one point - the center of the circumcircle.
So, we looked at the first remarkable point of the triangle - the point of intersection of its bisectoral perpendiculars.
Let's move on to the property of an arbitrary angle (see Fig. 4).
The angle is given, its bisector is AL, point M lies on the bisector.
Rice. 4
If point M lies on the bisector of an angle, then it is equidistant from the sides of the angle, that is, the distances from point M to AC and to BC of the sides of the angle are equal.
Proof:
Consider triangles and . These are right triangles and they are equal because... have a common hypotenuse AM, and the angles are equal, since AL is the bisector of the angle. Thus, right triangles are equal in hypotenuse and acute angle, it follows that , which is what needed to be proved. Thus, a point on the bisector of an angle is equidistant from the sides of that angle.
The converse theorem is true.
Theorem
If a point is equidistant from the sides of an undeveloped angle, then it lies on its bisector (see Fig. 5).
An undeveloped angle is given, point M, such that the distance from it to the sides of the angle is the same.
Prove that point M lies on the bisector of the angle.
Rice. 5
Proof:
The distance from a point to a line is the length of the perpendicular. From point M we draw perpendiculars MK to side AB and MR to side AC.
Consider triangles and . These are right triangles and they are equal because... have a common hypotenuse AM, legs MK and MR are equal by condition. Thus, right triangles are equal in hypotenuse and leg. From the equality of triangles follows the equality of the corresponding elements; equal angles lie opposite equal sides, thus, 
Therefore, point M lies on the bisector of the given angle.
If you need to inscribe a circle in an angle, this can be done, and there are infinitely many such circles, but their centers lie on the bisector of a given angle.
They say that a bisector is the locus of points equidistant from the sides of an angle.
A triangle consists of three angles. Let's construct the bisectors of two of them and get the point O of their intersection (see Fig. 6).
Point O lies on the bisector of the angle, which means it is equidistant from its sides AB and BC, let’s denote the distance as r: . Also, point O lies on the bisector of the angle, which means it is equidistant from its sides AC and BC: , , from here.
It is easy to notice that the point of intersection of the bisectors is equidistant from the sides of the third angle, which means it lies on
Rice. 6
angle bisector. Thus, all three bisectors of the triangle intersect at one point.
So, we remembered the proof of another important theorem.
The bisectors of the angles of a triangle intersect at one point - the center of the inscribed circle.
So, we looked at the second remarkable point of the triangle - the point of intersection of the bisectors.
We examined the bisector of an angle and noted its important properties: the points of the bisector are equidistant from the sides of the angle, in addition, the tangent segments drawn to the circle from one point are equal.
Let us introduce some notation (see Fig. 7).
Let us denote equal tangent segments by x, y and z. The side BC lying opposite the vertex A is designated as a, similarly AC as b, AB as c.
Rice. 7
Problem 1: in a triangle, the semi-perimeter and length of side a are known. Find the length of the tangent drawn from the vertex A - AK, denoted by x.
Obviously, the triangle is not completely defined, and there are many such triangles, but it turns out that they have some elements in common.
For problems involving an inscribed circle, the following solution method can be proposed:
1. Draw bisectors and get the center of the inscribed circle.
2. From center O, draw perpendiculars to the sides and obtain points of tangency.
3. Mark equal tangents.
4. Write out the relationship between the sides of the triangle and the tangents.
Goals:
- summarize students’ knowledge on the topic “Four remarkable points of a triangle”, continue work on developing skills in constructing the height, median, bisector of a triangle;
Introduce students to new concepts of the inscribed circle in a triangle and circumscribed around it;
Develop research skills;
- cultivate persistence, accuracy, and organization in students.
Task: expand cognitive interest in the subject of geometry.
Equipment: board, drawing tools, colored pencils, model of a triangle on a landscape sheet; computer, multimedia projector, screen.
During the classes
1.
Organizational moment (1 minute)
Teacher: In this lesson, each of you will feel like a research engineer; after completing the practical work, you will be able to evaluate yourself. For the work to be successful, it is necessary to carry out all actions with the model during the lesson very accurately and in an organized manner. I wish you success.
2.
Teacher: draw an open angle in your notebook
Q. What methods do you know of constructing the bisector of an angle?
Determination of the bisector of an angle. Two students construct angle bisectors on the board (using pre-prepared models) in two ways: with a ruler or compass. The following two students verbally prove the statements:
1. What properties do the bisector points of an angle have?
2. What can be said about the points lying inside the angle and equidistant from the sides of the angle?
Teacher: draw a tetragonal triangle ABC and in any of the ways, construct the bisectors of angle A and angle C, their point
intersection - point O. What hypothesis can you put forward about the ray VO? Prove that ray BO is the bisector of triangle ABC. Formulate a conclusion about the location of all bisectors of a triangle.
3.
Working with the triangle model (5-7 minutes).
Option 1 - acute triangle;
Option 2 - right triangle;
Option 3 - obtuse triangle.
Teacher: on the triangle model, construct two bisectors, circle them in yellow. Mark the point of intersection
bisector point K. See slide No. 1.
4.
Preparation for the main stage of the lesson (10-13 minutes).
Teacher: draw line segment AB in your notebook. What tools can be used to construct a perpendicular bisector to a segment? Determination of the perpendicular bisector. Two students are constructing a perpendicular bisector on the board
(according to pre-prepared models) in two ways: with a ruler, with a compass. The following two students verbally prove the statements:
1. What properties do the points of the perpendicular bisector to a segment have?
2. What can be said about the points equidistant from the ends of the segment AB? Teacher: draw a right triangle ABC in your notebook and construct the perpendicular bisectors to any two sides of the triangle ABC.
Mark the intersection point O. Draw a perpendicular to the third side through point O. What do you notice? Prove that this is the perpendicular bisector of the segment.
5.
Working with a triangle model (5 minutes).Teacher: on a triangle model, construct bisectoral perpendiculars to the two sides of the triangle and circle them in green. Mark the point of intersection of the bisectoral perpendiculars with a point O. See slide No. 2.
6.
Preparation for the main stage of the lesson (5-7 minutes).Teacher: draw an obtuse triangle ABC and construct two heights. Label their intersection point O.
1. What can be said about the third height (the third height, if extended beyond the base, will pass through point O)?
2. How to prove that all heights intersect at one point?
3. What new figure do these heights form and what are they in it?
7.
Working with the triangle model (5 minutes).
Teacher: on the triangle model, construct three heights and circle them in blue. Mark the point where the heights intersect with point H. See slide No. 3.
Lesson two
8.
Preparation for the main stage of the lesson (10-12 minutes).
Teacher: draw an acute triangle ABC and construct all its medians. Label their point of intersection O. What property do the medians of a triangle have?
9.
Working with the triangle model (5 minutes).
Teacher: on the triangle model, construct three medians and circle them in brown.
Mark the point of intersection of the medians with a point T. See slide No. 4.
10.
Checking the correctness of the construction (10-15 minutes).
1. What can be said about point K? / Point K is the point of intersection of bisectors, it is equidistant from all sides of the triangle /
2. Show on the model the distance from point K to the half side of the triangle. What shape did you draw? How is this located
cut to side? Highlight boldly with a simple pencil. (See slide number 5).
3. What is a point equidistant from three points of the plane that do not lie on the same straight line? Use a yellow pencil to draw a circle with center K and a radius equal to the distance marked with a simple pencil. (See slide number 6).
4. What did you notice? How is this circle located relative to the triangle? You have inscribed a circle in a triangle. What can you call such a circle?
The teacher gives the definition of an inscribed circle in a triangle.
5. What can be said about point O? \Point O is the point of intersection of the perpendicular bisectors and it is equidistant from all the vertices of the triangle\. What figure can be constructed by connecting points A, B, C and O?
6. Construct a circle (O; OA) using green. (See slide number 7).
7. What did you notice? How is this circle located relative to the triangle? What can you call such a circle? How can we call the triangle in this case?
The teacher gives the definition of a circumscribed circle around a triangle.
8. Attach a ruler to points O, H and T and draw a red line through these points. This line is called straight
Euler. (See slide number 8).
9. Compare OT and TN. Check FROM:TN=1: 2. (See slide number 9).
10. a) Find the medians of the triangle (in brown). Mark the bases of the medians with ink.
Where are these three points?
b) Find the altitudes of the triangle (in blue). Mark the bases of the heights with ink. How many of these points are there? \ Option 1-3; 2 option-2; Option 3-3\.c) Measure the distance from the vertices to the point of intersection of the heights. Name these distances (AN,
VN, SN). Find the midpoints of these segments and highlight them with ink. How many of these
points? \1 option-3; 2 option-2; Option 3-3\.
11. Count how many dots are marked with ink? \ 1 option - 9; Option 2-5; Option 3-9\. Designate
points D 1, D 2,…, D 9. (See slide number 10). Using these points you can construct an Euler circle. The center of the circle, point E, is in the middle of the segment OH. We draw a circle (E; ED 1) in red. This circle, like the straight line, is named after the great scientist. (See slide number 11).
11.
Presentation about Euler (5 minutes).
12. Summary(3 minutes). Score: “5” - if you get exactly the yellow, green and red circles and the Euler straight line. “4” - if the circles are 2-3mm inaccurate. “3” - if the circles are 5-7mm inaccurate.