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The inradius of the incircle in a triangle with sides of length , , is given by [7] = ... The large triangle is composed of six such triangles and the total area is: ...
Therefore, the Kepler triangle can be defined as the right triangle that, among all right triangles with the same hypotenuse, forms with its reflection the isosceles triangle of maximum inradius. [16] The same reflection also forms an isosceles triangle that, for a given perimeter, contains the largest possible semicircle. [17]
A triangle with sides <, semiperimeter = (+ +), area, altitude opposite the longest side, circumradius, inradius, exradii,, tangent to ,, respectively, and medians,, is a right triangle if and only if any one of the statements in the following six categories is true. Each of them is thus also a property of any right triangle.
In a right triangle two of the squares coincide and have a vertex at the triangle's right angle, so a right triangle has only two distinct inscribed squares. An obtuse triangle has only one inscribed square, with a side coinciding with part of the triangle's longest side.
In geometry, Euler's theorem states that the distance d between the circumcenter and incenter of a triangle is given by [1] [2] = or equivalently + + =, where and denote the circumradius and inradius respectively (the radii of the circumscribed circle and inscribed circle respectively).
where r is the inradius and R is the circumradius of the triangle. Here the sign of the distances is taken to be negative if and only if the open line segment DX (X = F, G, H) lies completely outside the triangle. In the diagram, DF is negative and both DG and DH are positive. The theorem is named after Lazare Carnot (1753–1823).
In a right triangle, the radius of the excircle on the hypotenuse equals the semiperimeter. The semiperimeter is the sum of the inradius and twice the circumradius. The area of the right triangle is () where a, b are the legs.
The inradius and circumradius formulas for an isosceles triangle may be derived from their formulas for arbitrary triangles. [31] The radius of the inscribed circle of an isosceles triangle with side length a {\displaystyle a} , base b {\displaystyle b} , and height h {\displaystyle h} is: [ 17 ]