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Some(?) of the Parts

A circle touches the lines OA, OB and AB where OA and OB are perpendicular. Show that the diameter of the circle is equal to the perimeter of the triangle


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?


M is any point on the line AB. Squares of side length AM and MB are constructed and their circumcircles intersect at P (and M). Prove that the lines AD and BE produced pass through P.


Age 14 to 16
Challenge Level

Square with a random point selected inside

Can you prove that the sum of the distances of any point inside a square from its sides is always equal?

Can you prove it to be true for a rectangle or a regular hexagon?
Does the hexagon need to be regular?

Can you show the same is the case for a regular pentagon?
Does the pentagon need to be regular?