Dilating the Ortholateral
- Paul Miller
The ortholateral ('orthogonal-quadriliateral') see Connecting the lines... combines ideas from chromogeometry (CG) and projective geometry (PG).
A quadrilateral (an object from PG) possesses a naturally associated object in CG when taking orthocentres (blue, red and green) of the four triangles formed from it. By the Gauss-Bodenmiller Theorem, orthocentres lies on a line (the GBm line - but more usually, 'radical-axis') which is perpendicular to a second line joining the midpoints of the line seqments completing the quadrilateral. CG introduces two further lines (through its additional perpendicular definitions) forming three mutually perpedicular lines. The result is a quadrilateral composed only of mutually perpendicular lines so the GBm lines (now threefold) ARE the lines completing the quadrilateral and the use of CG has brought the classical non-metrical PG construction into alignment with a metrical object - namely the ortho(gonal quadri)lateral or OL. The OL (as a quadrilateral) possesses its own ortholateral - and by extension its own 'anti'-ortholateral also.