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 perpendicular.

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?

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. . . .

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?

Can you decide whether two lines are perpendicular or not? Can you do this without drawing them?

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.

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?

Given any three non intersecting circles in the plane find another circle or straight line which cuts all three circles orthogonally.

Position the lines so that they are perpendicular to each other. What can you say about the equations of perpendicular lines?

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 ?