(See also ‘considerations‘, a follow-up blog from December 2019)
AN ortholateral is a(n orthogonal) quadrilateral formed of two pairs of perpendicular lines (crosses) in either the ‘blue’, ‘red’ or ‘green’ sense of ‘chromogeometry‘ (video clip). If two such ‘blue’ crosses are thus inclined in either the ‘green’ or the ‘red’ sense their lines commute into two distinct ‘green’ (or ‘red’) crosses that meet in a ‘blue’ sense.
AN equivalent description that produces this construction is to take an initial line, reflect it in a red (or green) ‘null-line’, reflect this combination in a green (or red) alternate null line and translate at least one line parallel to itself by an arbitrary amount, which could include no translation. (The apparent dependence of the ortholateral on Euclidean (blue) geometry is somewhat an illusion therefore.)
[ANOTHER equivalent description comes from looking at a suitably oriented triangle and its altitudes (six lines, perpendicular in pairs.) The ortholateral is then one altitude plus the associated side from the triangle plus the opposite side of its orthic triangle and the perpendicular bisector of that side:
side, altitude, [orthic] side, perpendicular bisector]
Complete Quadrilateral, Radical axis, mutual perpendicularity
BY the Gauss-Bodenmiller Theorem[i], the midpoints of the three ‘diagonal’ segments completing a general quadrilateral are collinear, and (blue) circles whose diameters lie on these segments meet in precisely two points forming a radical-axis (blue) perpendicular to the medial line joining their centres. For an ortholateral (which is not the case with a general quadrilateral) however, the radical-axis aligns with the segment itself which joins the blue perpendiculars (‘crosses’). And in an ortholateral, with a triple form of perpendicularity present, this implies the remaining joins of crosses are respectively red and green perpendicular to the medial line and, transitively, that opposite vertices in the ortholateral form a mutually orthogonal set (a triply-right triangle in the language of chromogeometry.) Together with the medial line, therefore, all three joins of vertices forms a NEW ortholateral constructed on an existing one.
HOWEVER, also by the Gauss-Bodenmiller[ii] Theorem also, the four (blue) orthocentres of the four triangles associated with a general quadrilateral are collinear and lie on the same (blue) radical-axis. In chromogeometry, with its three distinct perpendiculars, there will be three sets of collinear orthocentres forming three radical-axes mutally ‘perpendicular’ to the medial line. The Theorem thus establishes that every quadrilateral has an associated ortholateral and, since every ortholateral forms a new ortholateral, we naturally consider the ‘anti-ortholateral’ relation (c/f the anti-orthic triangle) to a given ortholateral.
In search of an ‘anti-ortholateral’:
Projective geometry, Harmonic ratio, Dual constructions:
IN obtaining the ortholateral-of-ortholateral we drew lines completing the quadrilateral and formed a triply right triangle of diagonals*, plus the medial line. That being a general method in projective geometry, the vertices of an ortholateral must be in an harmonic ratio with the ‘diameters’ of its associated anti-ortholateral (which indicates a method for obtaining the latter’s construction).
TWO of the three sides of a triangle of diagonals in a quadrilateral cut off an interval along the third side which, for an ortholateral due to the presence of perpenducality of sides, can be divided harmonically (per the construction for the circles of Apollonius) by a blue circle centred at the midpoint of a ‘red’ vertex and a ‘green’ vertex. (In other words, the interval to be divided will be the side of the triangle of diagonals, with the associated circle diameter being the side of the ortholateral on which it lies.)
THE circle along one side is met by the line blue-perpendicular to it in two symmetric points. In addition, a ‘dual’ interval forms where the same lines, with their perpendicularity swapped, cross and meet the alternate side of the ortholateral and a circle with diameter on this side meets the line perpendicular to it in two new points. A third circle centred on the meet of both sides – at the acute ‘corner’ of the ortholateral – will pass through all four points and will meet the lines forming the sides of the ortholateral at another four points along its own diameters.
THE ortholateral has generated a total of four chords on this circle – two as diameters and two as perpendicular chords (which commute to oblique chords) giving not one but two cyclic quadrangles (four points in general position) which each allow for four separate lines, forming an anti-orthoilateral. We thus have a new result: every ortholateral is associated with a pair of ‘dual’ figures (c/f four anti-orthic triangles of a triangle being considered as an orthocentric system ) and to each ortholateral there is a ‘dual’.
BY analogy with triangle geometry, the ortholateral may be considered a system of four ‘ortho-lines’, each side respectively perpendicular to its complement of three lines (which form ‘trilaterals‘ rather than ‘triangles‘), the medial line playing the role of the orthocentre in an orthic triangle (i.e. not the orthocentre of the triangle) that completes the ‘ortholineal’ system.
THE perspectivity of a triangle with its orthic triangle now becomes dual in the sense of Desargue’s Theorem: any three lines of the ortholateral are side-perspective to the lines of the triangle of diagonals in its fourth line (representing an orthocentre). The orthic-of-orthic perspector is represented differently, by a side-perspectivity of the lines of the ortholateral representing the triangle of diagonals to the triangle of diagonals forming part of the ortholateral-of-ortholateral.
THIS brief description of a new geometrical construction opens up an exciting set of connections between classical (Euclidean) triangle geometry with projective geometry via the (new) development of quadratic-metrical geometry known as ‘chromogeometry’
*Note the ‘triangle of diagonals’ of a quadrilateral here corresponds to the ‘diagonal triangle’ of the dual quadrangle in projective geometry
Further note There is a projective dual quadrangle which can be constructed from an orthlolateral (per quadrilateral). Although the figure (when constructed) is not in general orthogonal, we speculate that it could be rectifiable by a suitable change of ‘viewing angle’ (amounting to a linear transformation) so as to become an orthogonal system in four points in either the blue red or green sense, depending on whether the set is ‘covex’ (red or green) or ‘concave (blue). (More to follow)