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Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

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DOTS Division

Take any pair of two digit numbers x=ab and y=cd where, without loss of generality, ab > cd . Form two 4 digit numbers r=abcd and s=cdab and calculate: {r^2 - s^2} /{x^2 - y^2}.

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Investigate sequences given by $a_n = \frac{1+a_{n-1}}{a_{n-2}}$ for different choices of the first two terms. Make a conjecture about the behaviour of these sequences. Can you prove your conjecture?

Quad in Quad

Age 14 to 16 Challenge Level:
Draw any convex quadrilateral and find the midpoints of the four edges. Join the midpoints to form a new quadrilateral.

Try it a few times starting with different convex quadrilaterals. You may like to use the GeoGebra interactivity to explore.

What do you notice about the new quadrilaterals formed by joining the midpoints?
Can you make a conjecture?
Can you prove it?

Is the area of PQRS always the same fraction of the area of ABCD? Can you explain why?

What if ABCD is a concave quadrilateral?