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Triangles
Ancient Greek mathematician Euclid defined three types of triangle according to the lengths of their sides:[1][2]
Greek: τῶν δὲ τριπλεύρων σχημάτων ἰσόπλευρον μὲν τρίγωνόν ἐστι τὸ τὰς τρεῖς ἴσας ἔχον πλευράς, ἰσοσκελὲς δὲ τὸ τὰς δύο μόνας ἴσας ἔχον πλευράς, σκαληνὸν δὲ τὸ τὰς τρεῖς ἀνίσους ἔχον πλευράς, lit. 'Of trilateral figures, an isopleuron [equilateral] triangle is that which has its three sides equal, an isosceles that which has two of its sides alone equal, and a scalene that which has its three sides unequal.'[3]
- An equilateral triangle (Greek: ἰσόπλευρον, romanized: isópleuron, lit. 'equal sides') has three sides of the same length. An equilateral triangle is also a regular polygon with all angles measuring 60°.[4]
- An isosceles triangle (Greek: ἰσοσκελὲς, romanized: isoskelés, lit. 'equal legs') has two sides of equal length.[note 1][5] An isosceles triangle also has two angles of the same measure, namely the angles opposite to the two sides of the same length. This fact is the content of the isosceles triangle theorem, which was known by Euclid. Some mathematicians define an isosceles triangle to have exactly two equal sides, whereas others define an isosceles triangle as one with at least two equal sides.[5] The latter definition would make all equilateral triangles isosceles triangles. The 45–45–90 right triangle, which appears in the tetrakis square tiling, is isosceles.
- A scalene triangle (Greek: σκαληνὸν, romanized: skalinón, lit. 'unequal') has all its sides of different lengths.[6] Equivalently, it has all angles of different measure.
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Equilateral Triangle
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Isosceles triangle
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Scalene triangle
Hatch marks, also called tick marks, are used in diagrams of triangles and other geometric figures to identify sides of equal lengths. A side can be marked with a pattern of "ticks", short line segments in the form of tally marks; two sides have equal lengths if they are both marked with the same pattern. In a triangle, the pattern is usually no more than 3 ticks. An equilateral triangle has the same pattern on all 3 sides, an isosceles triangle has the same pattern on just 2 sides, and a scalene triangle has different patterns on all sides since no sides are equal.
Similarly, patterns of 1, 2, or 3 concentric arcs inside the angles are used to indicate equal angles: an equilateral triangle has the same pattern on all 3 angles, an isosceles triangle has the same pattern on just 2 angles, and a scalene triangle has different patterns on all angles, since no angles are equal.
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