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The product of the incircle radius and the circumcircle radius of a triangle with sides , , and is [13] = (+ +). Some relations among the sides, incircle radius, and circumcircle radius are: [ 14 ] a b + b c + c a = s 2 + ( 4 R + r ) r , a 2 + b 2 + c 2 = 2 s 2 − 2 ( 4 R + r ) r . {\displaystyle {\begin{aligned}ab+bc+ca&=s^{2}+(4R+r)r,\\a^{2 ...
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 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).
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).
In trigonometry, the law of sines, sine law, sine formula, or sine rule is an equation relating the lengths of the sides of any triangle to the sines of its angles. According to the law, = = =, where a, b, and c are the lengths of the sides of a triangle, and α, β, and γ are the opposite angles (see figure 2), while R is the radius of the triangle's circumcircle.
Hence, given the radius, r, center, P c, a point on the circle, P 0 and a unit normal of the plane containing the circle, ^, one parametric equation of the circle starting from the point P 0 and proceeding in a positively oriented (i.e., right-handed) sense about ^ is the following:
Two triangles are said to be poristic triangles if they have the same incircle and circumcircle. Given a circle with Center O and radius R and another circle with center I and radius r, there are an infinite number of triangles ABC with Circle O(R) as circumcircle and I(r) as incircle if and only if OI 2 = R 2 − 2Rr. These triangles form a ...
In the figure at right, given circle k with centre O and the point P outside k, bisect OP at H and draw the circle of radius OH with centre H. OP is a diameter of this circle, so the triangles connecting OP to the points T and T′ where the circles intersect are both right triangles.