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It follows from this formula that, for any two inscribed squares in a triangle, the square that lies on the longer side of the triangle will have smaller area. [5] In an acute triangle, the three inscribed squares have side lengths that are all within a factor of 2 3 2 ≈ 0.94 {\displaystyle {\frac {2}{3}}{\sqrt {2}}\approx 0.94} of each other.
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 connection with the inscribed square problem, Eggleston (1958) observed that the Reuleaux triangle provides an example of a constant-width shape in which no regular polygon with more than four sides can be inscribed, except the regular hexagon, and he described a small modification to this shape that preserves its constant width but also ...
One may ask whether other shapes can be inscribed into an arbitrary Jordan curve. It is known that for any triangle and Jordan curve , there is a triangle similar to and inscribed in . [12] [13] Moreover, the set of the vertices of such triangles is dense in . [14]
A curvilinear triangle is a shape with three curved sides, for instance, a circular triangle with circular-arc sides. (This article is about straight-sided triangles in Euclidean geometry, except where otherwise noted.) Triangles are classified into different types based on their angles and the lengths of their sides.
The orthic triangle, with vertices at the base points of the altitudes of the given triangle, has the smallest perimeter of all triangles inscribed into an acute triangle, hence it is the solution of Fagnano's problem.
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]