A triangle PQR, right angled at P, slides on a horizontal floor
with Q and R in contact with perpendicular walls. What is the locus
Four rods are hinged at their ends to form a convex quadrilateral.
Investigate the different shapes that the quadrilateral can take.
Be patient this problem may be slow to load.
Four rods are hinged at their ends to form a quadrilateral with
fixed side lengths. Show that the quadrilateral has a maximum area
when it is cyclic.
Thank you to Mayank, Campion School, Bhopal,
India; Yung, from Hong Kong and Ruth from Manchester High School
for Girls for your solutions. Here is Ruth's solution:
Chi Kin, St Dominic's International School of
Lisbon, also gave an excellent proof including a discussion of the
degenerate cases where one vertex, A, B, C or D is moved on top of
another or on top of the point X. In these cases there is no longer
a convex quadrilateral ABCD.
For instance, if we move C on top of B, both the points B, C and
X are joined at one point. As a result, the circumcircle of the
triangle BXC is reduced to a point, and only 3 circles will be
left. Quadrilateral PQRS cannot be formed any more.
Consider also, as we move C on top of X, the common chord CX
will be eventually reduced to a point, and RQ will therefore
disappear. Quadrilateral PQRS can't be formed.
Finally, if we move C on top of A, two of the circles will be
overlapping the other two. As a result, RQ will also overlap PS,
and the quadrilateral PQRS is reduced to a line.