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An inscribed square can cover at most half the area of the triangle it is inscribed into. [3] It is exactly half when the triangle has a side whose altitude (the perpendicular distance from the side to the opposite vertex) equals the length of the side, and when the square is inscribed with its edge on this side of the triangle.
The Nagel triangle or extouch triangle of is denoted by the vertices , , and that are the three points where the excircles touch the reference and where is opposite of , etc. This T A T B T C {\displaystyle \triangle T_{A}T_{B}T_{C}} is also known as the extouch triangle of A B C {\displaystyle \triangle ABC} .
Inscribed circles of various polygons An inscribed triangle of a circle A tetrahedron (red) inscribed in a cube (yellow) which is, in turn, inscribed in a rhombic triacontahedron (grey). (Click here for rotating model) In geometry, an inscribed planar shape or solid is one that is enclosed by and "fits snugly" inside another geometric shape or ...
In geometry, a triangle center or triangle centre is a point in the triangle's plane that is in some sense in the middle of the triangle. For example, the centroid , circumcenter , incenter and orthocenter were familiar to the ancient Greeks , and can be obtained by simple constructions .
Triangles have many types based on the length of the sides and the angles. A triangle whose sides are all the same length is an equilateral triangle, [3] a triangle with two sides having the same length is an isosceles triangle, [4] [a] and a triangle with three different-length sides is a scalene triangle. [7]
The extended sides of the orthic triangle meet the opposite extended sides of its reference triangle at three collinear points. [23] [24] [22] In any acute triangle, the inscribed triangle with the smallest perimeter is the orthic triangle. [25] This is the solution to Fagnano's problem, posed in 1775. [26]
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.
The foci of the Steiner inellipse of a triangle are the intersections of the inellipse's major axis and the circle with center on the minor axis and going through the Fermat points. [7]: Thm. 6 As with any ellipse inscribed in a triangle ABC, letting the foci be P and Q we have [8]
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