A quadrilateral changes shape with the edge lengths constant. Show
the scalar product of the diagonals is constant. If the diagonals
are perpendicular in one position are they always perpendicular?
Plane 1 contains points A, B and C and plane 2 contains points A
and B. Find all the points on plane 2 such that the two planes are
Take any point P inside an equilateral triangle. Draw PA, PB and PC
from P perpendicular to the sides of the triangle where A, B and C
are points on the sides. Prove that PA + PB + PC is a constant.
What is the shortest distance through the middle of a dodecahedron between the centres of two opposite faces?
Find the area of the shaded region created by the two overlapping
triangles in terms of a and b?
It is impossible to trisect an angle using only ruler and compasses
but it can be done using a carpenter's square.
X is a moveable point on the hypotenuse, and P and Q are the feet
of the perpendiculars from X to the sides of a right angled
triangle. What position of X makes the length of PQ a minimum?
The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and. . . .
Given any three non intersecting circles in the plane find another
circle or straight line which cuts all three circles orthogonally.
The circumcentres of four triangles are joined to form a
quadrilateral. What do you notice about this quadrilateral as the
dynamic image changes? Can you prove your conjecture?
ABCD is a square. P is the midpoint of AB and is joined to C. A line from D perpendicular to PC meets the line at the point Q. Prove AQ = AD.
P is a point inside a square ABCD such that PA= 1, PB = 2 and PC =
3. How big is angle APB ?