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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

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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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Circumspection

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.

Two Regular Polygons

Age 14 to 16 Challenge Level:

Simon from Elizabeth College, Guernsey found solutions using a spreadsheet :

We have made up a spreadsheet from Simon's description, the columns are :

  • number of sides for Polygon A

  • exterior angle for Polygon A is 360/n

  • exterior angle for Polygon B is 81 - column B

  • number of sides required for that exterior angle is 360/column C

Here's the spreadsheet file : Two Regular Polygons

Simon continues :

Polygon A starts with few sides and increases, Polygon B is calculated for each case.

As A increases in side number, B must decrease its own side number.

So once A exceeds B no new solutions will be found.

For a target angle of 81 I found two solutions : 5-40 and 8-10.

I then tried it with a target angle of 54

This gave me four solutions : 7-140, 8-40, 10-20 and 12-15

Finally I tried with a target angle of 27

This gave me five solutions : 14-280, 15-120, 16-80, 20-40 and 24-30

Simon then summarised the situation:

As the number of sides doubles, the angle is halved, and in general : the number of sides is in inverse proportion to the exterior angle.

27 is one third of 81 so it includes solutions that are three times the solutions for 81.

54 is double 27 so it has solutions which are half the size of solutions for 27 provided that value is an integer.

Excellent thinking and skilful application of a spreadsheet Simon, very well done.