Triangle Inequality Proof Using Euclidean Geometry

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In summary, Euclidean geometry is a branch of mathematics that focuses on the study of geometric shapes and their properties, based on the work of the ancient Greek mathematician Euclid. It is characterized by five postulates, including the concept of straight lines, points, and circles, as well as the transitive property of congruence and the parallel postulate. Non-Euclidean geometry, on the other hand, explores the properties of curved surfaces and three-dimensional shapes and challenges the parallel postulate. Euclidean geometry has many practical applications in everyday life, such as in architecture, engineering, and design. Some famous theorems in Euclidean geometry include the Pythagorean theorem, the angle-sum theorem, and the laws of cosines
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ilaneden
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proof the following using only euclidean geometry:
Let S be any point inside a triangle ABC and let SP; SQ; SR be
perpendicular to the sides BC;CA;AB respectively, then
SA + SB + SC >= 2 (SP + SQ + SR)
Hint: Set P1; P2 be the feet of the perpendiculars from R and Q upon
BC. Prove fir st that (i) QR >= P1P2 and (ii) PRP1 and SBR are
similar triangles.
 
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  • #2
This is a standard inequality , named after Erdo:s.
 

1. What is Euclidean geometry?

Euclidean geometry is a branch of mathematics that focuses on the study of geometric shapes and their properties. It is based on the work of the ancient Greek mathematician Euclid and is commonly referred to as "flat" or "classical" geometry.

2. What are the basic principles of Euclidean geometry?

The basic principles of Euclidean geometry are based on five postulates, including the concept of a straight line, a point, and the ability to draw a circle with any given center and radius. Other principles include the transitive property of congruence and the parallel postulate.

3. What is the difference between Euclidean and non-Euclidean geometry?

While Euclidean geometry deals with flat, two-dimensional shapes and their properties, non-Euclidean geometry explores the properties of curved surfaces and three-dimensional shapes. Non-Euclidean geometry also challenges the parallel postulate, leading to a different set of principles and theorems than those found in Euclidean geometry.

4. How is Euclidean geometry used in real life?

Euclidean geometry has many practical applications in everyday life, including architecture, engineering, and design. It is used to create accurate maps and blueprints, design buildings and structures, and even in computer graphics and video games.

5. What are some famous theorems in Euclidean geometry?

Some of the most well-known theorems in Euclidean geometry include the Pythagorean theorem, which relates the sides of a right triangle, and the angle-sum theorem, which states that the sum of the angles in a triangle is always 180 degrees. Other famous theorems include the law of cosines, the law of sines, and the parallel lines theorem.

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