Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-12-05T02:22:48.387Z Has data issue: false hasContentIssue false
  • English
  • Français

Pedal Circles and the Quadrangle

Published online by Cambridge University Press:  03 November 2016

H. V. Mallison
H. V. Mallison*
Affiliation:
University of Exeter
Get access Rights & Permissions [Opens in a new window]

Extract

In two articles in the Mathematical Gazette, Pedal Triangles and Pedal Circles, July 1919, and Some Properties Relative to a Tetrastigm, Oct. 1920, J. H. Lawlor gave proofs of a number of properties of the pedal circles of the four vertices of a plane quadrangle ABCD with respect to the triangles formed by the remaining three vertices. The articles concluded with the statement of ten results, the first nine of which were fairly easily deducible from those already proved, but No. (10) was given as a conjecture. This was that the common chord of the pedal circles of A and D bisects BH1 and CH2, where H1and H2 are the orthocentres of the triangles BAD and CAD, and hence that BH2, CH1 meet on the common chord of the pedal circles of A and D. It can be seen that the second result would imply the former, for if h is the rectangular hyperbola through ABCD, which is unique unless the points are orthocentric, and if O is the centre of h, then H1 and H2 also lie on h, and it is known that the pedal circles pass through O. The meet of BH2, CH1 is a diagonal point of the quadrangle BH1H2C inscribed in h, and its join to the centre of h is the diameter conjugate to the parallel chords BH1CH2 and therefore bisects them.

Type
Research Article
Information
The Mathematical Gazette , Volume 42 , Issue 339 , February 1958 , pp. 17 - 20
Copyright
Copyright © Mathematical Association 1958

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

* AQ, DP meet at H 1 and AS, DR meet at H 2. By applying Pascal’s theorem to the hexagons APSDQR, AQRDPS, AQSDPR, and ASQDRP it is seen that K,L are diagonal points of the quadrangle BH 1 H 2 G and hence KL bisects the parallel sides BH 1 CH 2.