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.

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?

At the time of writing the hour and minute hands of my clock are at right angles. How long will it be before they are at right angles again?

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

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.

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. . . .

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

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?