Hello everyone, today we will learn about concurrent lines in triangles.
There are basically four types of concurrent lines, which are: median line, elevated road, median line and bisectrix.
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It is not possible to fully introduce the above four types of lines in the same article, so I only state the definition, draw illustrations and present some basic knowledge that is directly related. stop.
The purpose of this article is very simple, when reading its name, you can draw it correctly. So that’s ok!
#first. What is the median of the triangle?
+ The median of a triangle is a line passing through the vertex and midpoint of the opposite side. Each triangle will have 3 medians.
The three medians in a triangle always converge at a point, this point is called the centroid of the triangle.
The centroid of the triangle is at a distance from the vertex that is two-thirds the length of the corresponding median.
AMis the median of triangle ABC, it passes through the vertexAand midpointMof edgeBCGis the centroid of the triangleABC- $AG=\frac{2}{3}AM$
+ In a right triangle then the length of the median corresponding to the hypotenuse is ½ of the length of the hypotenuse.
AM is the median of the right triangle ABC (square at A) so $AM=\frac{1}{2}BC$
+ In an isosceles triangletwo medians corresponding to two sides are congruent.
As shown above, BN and CM are the two medians of the isosceles triangle ABC (isosceles at A), so BN=CM
#2. What is the altitude of the triangle?
Altitude of the triangle is a line that passes through the vertex and is perpendicular to the opposite side.
In other words: The altitude of a triangle is the line segment drawn from a vertex and perpendicular to the opposite side of that triangle.
The three altitudes in a triangle always converge at a point, this point is called the orthocenter of the triangle.
AA’is the high road that goes through the topAand perpendicular to the opposite sideBCinA’Gis the orthocenter of the triangleABC
+ In an isosceles triangle The altitude passing through the vertex is also the median, the perpendicular bisector and the bisector of that triangle.
As shown above, the altitude AA’ passes through the vertex A of the isosceles triangle ABC (isosceles at A), so AA’ is also the median, the perpendicular bisector and the bisector.
+ In a triangle with two of the four types of lines (median, altitude, orthogonal, bisector) coincide, then the triangle is isosceles triangle => you can apply this knowledge to prove the triangle is assumed to be an isosceles triangle.
#3. What is the perpendicular bisector of the triangle?
The perpendicular bisector of a triangle is the line that passes through the midpoint of an side and is perpendicular to that side.
The three orthogonals in a triangle always converge at a point, which is equidistant from the three vertices of the triangle and is the centroid circumcircle of triangle.
OMis the perpendicular bisector of the sideBCOis the center of the circumcircle of the triangleABC- OA = OB = OC
+ In an isosceles triangle The median line that corresponds to the base edge is also the median.
As shown above, AA’ is the perpendicular bisector of the isosceles triangle ABC (isosceles at A), so it is also the median
#4. What is the bisector of the triangle?
Before coming to the concept of bisectors, we will learn about the concept of bisectors first.
The bisector of an angle is the ray that lies between the two sides of the angle and forms with them two equal angles.
The bisector of $\hat{A}$ intersects the side opposite BC at point A’, then AA’ is called the bisector of $\hat{A}$
The three bisectors in a triangle always converge at a point, this point is equidistant from the three sides of the triangle and is the center of the incircle of the triangle.
AA’is a bisector corresponding to the sideBCIis the center of the inscribed circle
#5. Epilogue
Well, so through this article, you already know or recall the properties and definitions of median line, elevated road, median line, bisectrix in a triangle, right?
- In an equilateral triangle then the centroid, orthocenter, circumcenter, and incircle center will coincide.
- In any triangle then the centroid, orthocenter, and circumcenter of the triangle always lie on a straight line, or in other words, are collinear.
Those are a couple of words of knowledge expansion that I want to send to you.
Hope this article will be useful to you, wish you a good study, goodbye and see you in the next articles!
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