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Do triangle inequalities always hold?
Triangle inequality, in Euclidean geometry, theorem that the sum of any two sides of a triangle is greater than or equal to the third side; in symbols, a + b ≥ c. With appropriate norms, the triangle inequality holds for complex numbers, integrals, and other abstract spaces in functional analysis. …
When triangle inequality is an equality?
In the Euclidean case, equality occurs only if the triangle has a 180° angle and two 0° angles, making the three vertices collinear, as shown in the bottom example. Thus, in Euclidean geometry, the shortest distance between two points is a straight line.
What is the triangle inequality rule?
The triangle inequality theorem simply states that the sum of any two sides in a triangle must be greater than the length of the third side, or else it isn’t a triangle.
How do you verify a triangle inequality?
The 3 properties of the triangle inequality theorem are:
- If the sum of any two sides is greater than the third, then the difference of any two sides will be less than the third.
- The sum of any two sides must be greater than the third side.
- The side opposite to a larger angle is the longest side in the triangle.
Can these 3 sides make a triangle?
Can three equal side lengths form a triangle? Yes. It’s called an equilateral triangle, and it can work because two side lengths added together are bigger than the third side.
Which is a defining property of the triangle inequality?
The triangle inequality is a defining property of norms and measures of distance.
How is the triangle inequality extended to polygonal paths?
When r the common ratio is chosen such that r = √φ it generates a right triangle that is always similar to the Kepler triangle . The triangle inequality can be extended by mathematical induction to arbitrary polygonal paths, showing that the total length of such a path is no less than the length of the straight line between its endpoints.
How did Euclid prove the triangle inequality for distances?
Euclid proved the triangle inequality for distances in plane geometry using the construction in the figure. Beginning with triangle ABC, an isosceles triangle is constructed with one side taken as BC and the other equal leg BD along the extension of side AB.
Is the triangle inequality a consequence of the law of cosines?
In Euclidean geometry, for right triangles the triangle inequality is a consequence of the Pythagorean theorem, and for general triangles a consequence of the law of cosines, although it may be proven without these theorems. The inequality can be viewed intuitively in either ℝ2 or ℝ3.