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The radius of the incircle is related to the area of the triangle. [18] The ratio of the area of the incircle to the area of the triangle is less than or equal to /, with equality holding only for equilateral triangles. [19]
A tangential polygon has a larger area than any other polygon with the same perimeter and the same interior angles in the same sequence. [ 6 ] : p. 862 [ 7 ] The centroid of any tangential polygon, the centroid of its boundary points, and the center of the inscribed circle are collinear , with the polygon's centroid between the others and twice ...
A tangential quadrilateral with its incircle. In Euclidean geometry, a tangential quadrilateral (sometimes just tangent quadrilateral) or circumscribed quadrilateral is a convex quadrilateral whose sides all can be tangent to a single circle within the quadrilateral.
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.
The area of a bicentric quadrilateral can be expressed in terms of two opposite sides and the angle θ between the diagonals according to [9] = = . In terms of two adjacent angles and the radius r of the incircle, the area is given by [9]
Harcourt's theorem is a formula in geometry for the area of a triangle, as a function of its side lengths and the perpendicular distances of its vertices from an arbitrary line tangent to its incircle. [1] The theorem is named after J. Harcourt, an Irish professor. [2]
This can be seen from the area formula πr 2 and the circumference formula 2πr. ... The ratio of the area of the incircle to the area of an equilateral triangle, ...
Using the usual notations for a triangle (see the figure at the upper right), where a, b, c are the lengths of the three sides, A, B, C are the vertices opposite those three respective sides, α, β, γ are the corresponding angles at those vertices, s is the semiperimeter, that is, s = a + b + c / 2 , and r is the radius of the inscribed circle, the law of cotangents states that