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The inradius of the incircle in a triangle with sides of length , , is given by [7] = ... This is a right-angled triangle with one side equal to ...
A right triangle ABC with its right angle at C, hypotenuse c, and legs a and b,. 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).
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]
Let I be the center of the incircle of triangle ABC, r its radius and F a, F b and F c the three points where the incircle touches the triangle sides a, b and c. Since the (extended) triangle sides are tangents of the incircle it follows that IF a, IF b and IF c are perpendicular to a, b and c.
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.
A triangle in which one of the angles is a right angle is a right triangle, ... Its radius is called the inradius. There are three other important circles, ...
Every angle of an integer right triangle also has rational sine (see Pythagorean triple). If the angles of any triangle form an arithmetic progression then one of its angles must be 60°. [5] For integer triangles the remaining angles must also have rational cosines and a method of generating such triangles is given below.
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.