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An equilateral triangle may have integer sides with three rational angles as measured in degrees, [13] known for the only acute triangle that is similar to its orthic triangle (with vertices at the feet of the altitudes), [14] and the only triangle whose Steiner inellipse is a circle (specifically, the incircle).
In the latter double inequality, the first part holds with equality if and only if the triangle is isosceles with an apex angle of at least 60°, and the last part holds with equality if and only if the triangle is isosceles with an apex angle of at most 60°. Thus both are equalities if and only if the triangle is equilateral. [7]: Thm. 1
The only integer triangle with three rational angles (rational numbers of degrees, or equivalently rational fractions of a full turn) is the equilateral triangle. [2] This is because integer sides imply three rational cosines by the law of cosines , and by Niven's theorem a rational cosine coincides with a rational angle if and only if the ...
Rewriting the inequality above allows for a more concrete geometric interpretation, which in turn provides an immediate proof. [1]+ +. Now the summands on the left side are the areas of equilateral triangles erected over the sides of the original triangle and hence the inequation states that the sum of areas of the equilateral triangles is always greater than or equal to threefold the area of ...
A triangle in which one of the angles is a right angle is a right triangle, a triangle in which all of its angles are less than that angle is an acute triangle, and a triangle in which one of it angles is greater than that angle is an obtuse triangle. [8] These definitions date back at least to Euclid. [9]
The equilateral triangle, with three 60° angles, is acute. The Morley triangle, formed from any triangle by the intersections of its adjacent angle trisectors, is equilateral and hence acute. The golden triangle is the isosceles triangle in which the ratio of the duplicated side to the base side equals the golden ratio. It is acute, with ...
Any two pairs of angles are congruent, [4] which in Euclidean geometry implies that all three angles are congruent: [a] If ∠BAC is equal in measure to ∠B'A'C', and ∠ABC is equal in measure to ∠A'B'C', then this implies that ∠ACB is equal in measure to ∠A'C'B' and the triangles are similar. All the corresponding sides are ...
If each vertex angle of the outer triangle is trisected, Morley's trisector theorem states that the purple triangle will be equilateral. In plane geometry, Morley's trisector theorem states that in any triangle, the three points of intersection of the adjacent angle trisectors form an equilateral triangle, called the first Morley triangle or simply the Morley triangle.