In elementary geometry, a

quadrilateral In Euclidean geometry, geometry a quadrilateral is a four-sided polygon, having four Edge (geometry), edges (sides) and four Vertex (geometry), corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''l ...

whose diagonals are perpendicular and of equal length has been called a midsquare quadrilateral (referring to the

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 ...

formed by its four edge

midpoint In geometry, the midpoint is the middle point of a line segment. It is equidistant from both endpoints, and it is the centroid both of the segment and of the endpoints. It bisects the segment. Formula The midpoint of a segment in ''n''-dim ...

s). These shapes are, by definition, simultaneously

equidiagonal quadrilateral In Euclidean geometry, an equidiagonal quadrilateral is a convex polygon, convex quadrilateral whose two diagonals have equal length. Equidiagonal quadrilaterals were important in ancient Indian mathematics, where quadrilaterals were classified fir ...

s and

orthodiagonal quadrilateral In Euclidean geometry, an orthodiagonal quadrilateral is a quadrilateral in which the diagonals cross at right angles. In other words, it is a four-sided figure in which the line segments between non-adjacent vertices are orthogonal (perpendicula ...

s. Older names for the same shape include pseudo-square, and skewsquare.

Midsquare

In any quadrilateral, the four edge midpoints form a parallelogram, the

Varignon parallelogram In Euclidean geometry, Varignon's theorem holds that the midpoints of the sides of an arbitrary quadrilateral form a parallelogram, called the Varignon parallelogram. It is named after Pierre Varignon, whose proof was published posthumously in 17 ...

, whose sides are parallel to the diagonals and half their length. It follows that, in an equidiagonal and orthodiagonal quadrilateral, the sides of the Varignon parallelogram are equal-length and perpendicular; that is, it is a square. For the same reason, a quadrilateral whose Varignon parallelogram is square must be equidiagonal and orthodiagonal. This characterization motivates the midsquare quadrilateral name for these shapes. A midsquare quadrilateral can be constructed from its midsquare and any one of its vertices. To do so, let the given midsquare have vertices

P, Q, R, S

, and let a vertex

A

of the midsquare quadrilateral be given. Then the remaining three vertices

B,C,D

can be constructed by letting

B

be the reflection of

A

across

P

(so that

P

is the midpoint of segment

AB

, letting

C

be the reflection of

B

across

Q

, and letting

D

be the reflection of

C

R

. It will automatically follow that

A

is the reflection of

D

across

S

, completing the midsquare quadrilateral.

Van Aubel and diagonal squares

According to Van Aubel's theorem, a midsquare quadrilateral can also be constructed from an arbitrary quadrilateral by placing squares on the four sides of the quadrilateral, exterior to it, and connecting the four centers of the squares. Skewsquare

For any two opposite sides of a midsquare quadrilateral, the two squares having these sides as their diagonals intersect in a single vertex, called a ''focus'' of the quadrilateral. Conversely, if two squares intersect in a vertex, then their two diagonals disjoint from this vertex form two opposite sides of a (possibly non-convex) midsquare quadrilateral. The fact that the resulting quadrilateral has a midsquare can be seen as an instance of the

Finsler–Hadwiger theorem The Finsler–Hadwiger theorem is statement in Euclidean plane geometry that describes a third square derived from any two squares that share a vertex. The theorem is named after Paul Finsler and Hugo Hadwiger, who published it in 1937 as part ...

. The two foci and the two diagonal midpoints of any midsquare quadrilateral form the vertices of a square. Each focus lies on an angle bisector of the two diagonals and on the perpendicular bisectors of the two sides that are the diagonals of its squares. The outer four vertices of the four diagonal squares of a midsquare quadrilateral form another midsquare quadrilateral. These are the same four points that would be obtained by applying Van Aubel's theorem to the given midsquare quadrilateral.

Biggest little quadrilateral

Midsquare quadrilaterals whose sides are not longer than the diagonals have the maximum

area Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...

for their

diameter In geometry, a diameter of a circle is any straight line segment that passes through the centre of the circle and whose endpoints lie on the circle. It can also be defined as the longest Chord (geometry), chord of the circle. Both definitions a ...

among all quadrilaterals, solving the

n=4

case of the biggest little polygon problem. The square is one such quadrilateral, but there are infinitely many others.

Examples

Midsquare quadrilateral.svg , example of a midsquare quadrilateral Midsquare trapezoid.svg , a midsquare

trapezoid In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides. The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...

Midsquare kite.svg , a midsquare

kite A kite is a tethered heavier than air flight, heavier-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. Kites often have ...

Midsquare square.svg , a midsquare

parallelogram In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

, that is, a square

References

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