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In an acute triangle, the sum of the circumradius R and the inradius r is less than half the sum of the shortest sides a and b: [4]: p.105, #2690. while the reverse inequality holds for an obtuse triangle. For an acute triangle with medians ma , mb , and mc and circumradius R, we have [4]: p.26, #954.
convex, cyclic. Dual polygon. Self-dual. In geometry, an isosceles triangle (/ aɪˈsɒsəliːz /) is a triangle that has two sides of equal length. Sometimes it is specified as having exactly two sides of equal length, and sometimes as having at least two sides of equal length, the latter version thus including the equilateral triangle as a ...
Angle. A green angle formed by two red rays on the Cartesian coordinate system. In Euclidean geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. [1] Angles formed by two rays are also known as plane angles as they lie in the plane that contains the rays.
A right triangle or right-angled triangle, sometimes called an orthogonal triangle or rectangular triangle, is a triangle in which two sides are perpendicular, forming a right angle (1⁄4 turn or 90 degrees). The side opposite to the right angle is called the hypotenuse (side in the figure). The sides adjacent to the right angle are called ...
An obtuse trapezoid on the other hand has one acute and one obtuse angle on each base. An isosceles trapezoid is a trapezoid where the base angles have the same measure. As a consequence the two legs are also of equal length and it has reflection symmetry. This is possible for acute trapezoids or right trapezoids (as rectangles).
Edsger W. Dijkstra has stated this proposition about acute, right, and obtuse triangles in this language: sgn(α + β − γ) = sgn(a 2 + b 2 − c 2), where α is the angle opposite to side a, β is the angle opposite to side b, γ is the angle opposite to side c, and sgn is the sign function. [30]
A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called vertices, are zero- dimensional points while the sides connecting them, also called edges, are one-dimensional line segments. A triangle has three internal angles, each one bounded by a pair of adjacent edges; the sum of ...
The change of sign as the angle γ becomes obtuse makes a case distinction necessary. Recall that a 2, b 2, and c 2 are the areas of the squares with sides a, b, and c, respectively; if γ is acute, then ab cos γ is the area of the parallelogram with sides a and b forming an angle of γ′ = π / 2 − γ;