Challenge
#endboxedheading564 Circle geometry (Chapter 27)1 A solid bar AB moves so that A remains on thex-axis
and B remains on they-axis. At P, the midpoint of AB,
is a small light.
Prove that as A and B move to all possible positions, the
light traces out a path which forms a circle.
(Do not use coordinate geometry methods.)2 PAB is a wooden set square in which AbPB is a right angle.
The set square is free to move so that A is always on the
x-axis and B is always on they-axis.
Show that the point P always lies on a straight line
segment which passes through O.
(Do not use coordinate geometry methods.)3 P is any point on the circumcircle of¢ABC other than
at A, B or C. Altitudes PX, PY and PZ are drawn to the
sides of¢ABC (or the sides produced).
Prove that X, Y and Z are collinear. XYZ is known as
Simson’s line.4 Britney notices that her angle of view of a picture on a
wall depends on how far she is standing in front of the
wall. When she is close to the wall the angle of view
is small. When she moves backwards so that she is a
long way from the wall the angle of view is also small.
It becomes clear to Britney that there must be a point
in the room where the angle of view is greatest. She
is wondering whether this position can be found from a
deductive geometry argument only. Kelly said that she
thought this could be done by drawing an appropriate
circle.
She said that the solution is to draw a circle through A
and B which touches the ‘eye level’ line at P, then AbPB is
the largest angle of view. To prove this, choose any other
point Q on the eye level line and show that this angle
must be less than AbPB. Complete the full argument.XYPAB C ZyA xPBO
yA xPBOeye
levelABangle
of vieweye
levelAB
P
QpictureIGCSE01
cyan magenta yellow black(^05255075950525507595)
100 100
(^05255075950525507595)
100 100
Y:\HAESE\IGCSE01\IG01_27\564IGCSE01_27.cdr Wednesday, 29 October 2008 1:46:26 PM PETER