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Structure Euler's Line

The document discusses concepts related to triangles including the circumcenter, orthocenter, and centroid of a triangle. It then defines the Euler Line as the line containing the centroid, circumcenter, and orthocenter, which are always collinear for any triangle. The document was named after Leonhard Euler, an influential 18th century Swiss mathematician who made important contributions to fields like calculus and graph theory and introduced modern mathematical notation. It then provides further details on these concepts and the Euler Line for different types of triangles like right, isosceles, equilateral, and scalene triangles.

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230 views2 pages

Structure Euler's Line

The document discusses concepts related to triangles including the circumcenter, orthocenter, and centroid of a triangle. It then defines the Euler Line as the line containing the centroid, circumcenter, and orthocenter, which are always collinear for any triangle. The document was named after Leonhard Euler, an influential 18th century Swiss mathematician who made important contributions to fields like calculus and graph theory and introduced modern mathematical notation. It then provides further details on these concepts and the Euler Line for different types of triangles like right, isosceles, equilateral, and scalene triangles.

Uploaded by

Study DB
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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Content

1. Visiting some important concepts


a. Circumcentre of a triangle
b. Orthocentre of a triangle
c. Centroid of a triangle
2. What is Euler Line?
a. The Mind Behind Euler Line
b. Defining the Euler Line
3. Geometrical Importance of Euler Line
4. Euler Lines in Special Triangles
a. Right-angled triangle
b. Isosceles triangle
c. Equilateral triangle
d. Scalene triangle

Visiting Some Important Concepts


Circumcentre of a Triangle:
For every triangle there is exactly one circle (called the circumcircle) that passes through the three
vertices of the triangle. Moreover, the centre of the circumcircle is the intersection of the three
perpendicular bisectors of the sides of the triangle; this centre is called the circumcentre of the
triangle. The circumcentre of an acute triangle lies inside the triangle; for an obtuse triangle its
circumcentre lies outside the triangle; while in the case of a right triangle its circumcentre is the
midpoint of the hypotenuse.

Orthocentre of a Triangle:
In any triangle, the three altitudes are concurrent. The common intersection point of the three
altitudes is called the orthocentre which lies inside acute triangles, outside obtuse triangles, and
coincides with the vertex of right angle in a right triangle.

Centroid of a Triangle:
The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex
with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1,
which is to say it is located ⅓ of the distance from each side to the opposite vertex. The centroid
always lies inside all triangles

What is Euler Line?


The Mind behind Euler Line:
Leonhard Euler (15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and
physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph
theory. He also introduced much of the modern mathematical terminology and notation, particularly
for mathematical analysis, such as the notion of a mathematical function: he was the first to write
f(x) to denote the function f applied to the argument x. He also introduced the modern notation for
the trigonometric functions, the letter e for the base of the natural logarithm (now also known as
Euler’s number), the Greek letter Σ for summations and the letter i to denote the imaginary unit. The
use of the Greek letter π to denote the ratio of a circle’s circumference to its diameter was also
popularized by Euler, although it did not originate with him. He is also renowned for his work in
mechanics, fluid dynamics, optics, and astronomy.

Defining the Euler Line:


In any triangle, three of the many centres of a triangle, the centroid, circumcentre and the
orthocentre are always collinear. These three points always lie on a straight line. This line is called
the Euler Line, named after the Swiss mathematician Leonhard Euler.

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