## (See also ‘considerations‘, a follow-up blog from December 2019)

## Introduction

#### Description:

**AN ortholateral** is a(n *ortho*gonal) quad*rilateral* 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

#### Collinear orthocentres:

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’.

### Discussion points:

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 ‘tri*laterals*‘ rather than ‘tri*angles*‘), 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)