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The area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.
In geometry, the trilinear coordinates x : y : z of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates .
The interior angles of an ideal triangle are all zero. An ideal triangle has infinite perimeter. An ideal triangle is the largest possible triangle in hyperbolic geometry. In the standard hyperbolic plane (a surface where the constant Gaussian curvature is −1) we also have the following properties: Any ideal triangle has area π. [1]
The Nagel point is the isotomic conjugate of the Gergonne point.The Nagel point, the centroid, and the incenter are collinear on a line called the Nagel line.The incenter is the Nagel point of the medial triangle; [2] [3] equivalently, the Nagel point is the incenter of the anticomplementary triangle.
Barycentric coordinates are strongly related with Cartesian coordinates and, more generally, to affine coordinates (see Affine space § Relationship between barycentric and affine coordinates). Barycentric coordinates are particularly useful in triangle geometry for studying properties that do not depend on the angles of the triangle, such as ...
A triangle with sides a, b, and c. In geometry, Heron's formula (or Hero's formula) gives the area of a triangle in terms of the three side lengths , , . Letting be the semiperimeter of the triangle, = (+ +), the area is [1]
The Cartesian coordinates of the incenter are a weighted average of the coordinates of the three vertices using the side lengths of the triangle relative to the perimeter—i.e., using the barycentric coordinates given above, normalized to sum to unity—as weights. (The weights are positive so the incenter lies inside the triangle as stated ...
The inellipse with the largest area is the Steiner inellipse, also called the midpoint inellipse, with its center at the triangle's centroid. [3]: p.145 In general, the ratio of the inellipse's area to the triangle's area, in terms of the unit-sum barycentric coordinates (α, β, γ) of the inellipse's center, is [3]: p.143