elementary geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, an inscribed square in a triangle is a

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

whose four vertices all lie on a given

triangle A triangle is a polygon with three corners and three sides, one of the basic shapes in geometry. The corners, also called ''vertices'', are zero-dimensional points while the sides connecting them, also called ''edges'', are one-dimension ...

. By the

pigeonhole principle In mathematics, the pigeonhole principle states that if items are put into containers, with , then at least one container must contain more than one item. For example, of three gloves, at least two must be right-handed or at least two must be l ...

, two of the square's vertices, and the edge between them, must lie on one of the sides of the triangle. For instance, for the Calabi triangle depicted, the square with horizontal and vertical sides is inscribed; the other two squares in the figure are not inscribed. This is a special case of the

inscribed square problem The inscribed square problem, also known as the square peg problem or the Toeplitz conjecture, is an unsolved question in geometry: ''Does every plane simple closed curve contain all four vertices of some square?'' This is true if the curve is ...

asking for a square whose vertices lie on a simple closed curve. However, although the inscribed square problem remains unsolved in general, it is known to have a solution for every

polygon In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon ...

and for every

convex set In geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube (geometry), cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is n ...

, two special cases that both apply to triangles. Every

acute triangle An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one ''obtuse angle'' (greater than 90°) and two acute angles. Since a triang ...

has three inscribed squares, one lying on each of its three sides. In a

right triangle 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 ( turn or 90 degrees). The side opposite to the right angle i ...

there are two inscribed squares, one touching the right angle of the triangle and the other lying on the opposite side. An

obtuse triangle An acute triangle (or acute-angled triangle) is a triangle with three ''acute angles'' (less than 90°). An obtuse triangle (or obtuse-angled triangle) is a triangle with one '' obtuse angle'' (greater than 90°) and two acute angles. Since a trian ...

has only one inscribed square, with a side coinciding with part of the triangle's longest side. The Calabi triangle, an obtuse triangle, shares with the

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

the property of having three different ways of placing the largest square that fits into it, but (because it is obtuse) only one of these three is inscribed. An inscribed square can cover at most half the area of the triangle it is inscribed into. It is exactly half when the triangle has a side whose

altitude Altitude is a distance measurement, usually in the vertical or "up" direction, between a reference datum (geodesy), datum and a point or object. The exact definition and reference datum varies according to the context (e.g., aviation, geometr ...

(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. In all other cases, the inscribed square is smaller than half the triangle. For a square that lies on a triangle side of length

s

, with altitude

h

, the square's side length will be

\frac.

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. In an acute triangle, the three inscribed squares have side lengths that are all within a factor of

\frac23\sqrt2\approx 0.94

of each other.

References

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Mathematics Magazine ''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a j ...

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Math Horizons ''Math Horizons'' is a magazine aimed at undergraduates interested in mathematics, published by the Mathematical Association of America. It publishes expository articles about "beautiful mathematics" as well as articles about the culture of mathem ...

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