Midsquare quadrilateral

In elementary geometry, a quadrilateral whose diagonals are perpendicular and of equal length has been called a midsquare quadrilateral (referring to the square formed by its four edge midpoints).

[2] Older names for the same shape include pseudo-square,[3][4] and skewsquare.

[4] In any quadrilateral, the four edge midpoints form a parallelogram, the Varignon parallelogram, 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.

[5] This characterization motivates the midsquare quadrilateral name for these shapes.

[2][6] 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.

[4][1] The fact that the resulting quadrilateral has a midsquare can be seen as an instance of the Finsler–Hadwiger theorem.

[7] 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.

These are the same four points that would be obtained by applying Van Aubel's theorem to the given midsquare quadrilateral.