G.574 circles center inscribe — Heath

XII.16

Given two circles about the same centre, to inscribe in the greater circle an equilateral polygon with an even number of sides which does not touch the lesser circle.

Let ( A B C D ), ( E F G H ) be the two given circles about the same centre K ; thus it is required to inscribe in the greater circle ( A B C D ) an equilateral polygon with an even number of sides which does not touch the circle ( E F G H ).

For let the straight line / B K D / be drawn through the centre K, and from the point G let / G A / be drawn at right angles to the straight line / B D / and carried through to C ;

[G.118]therefore / A C / touches the circle ( E F G H ).

Then, bisecting the circumference ( B A D ), bisecting the half of it, and doing this continually, we shall leave a circumference less than ( A D ). [G.371] Let such be left, and let it be ( L D ) ; from L let / L M / be drawn perpendicular to / B D / and carried through to N, and let / L D /, / D N / be joined ;

[G.100, G.24]therefore / L D / is equal to / D N /.

Now, since / L N / is parallel to / A C /, and / A C / touches the circle ( E F G H ),

therefore / L N / does not touch the circle ( E F G H ) ; therefore / L D /, / D N / are far from touching the circle ( E F G H ).

If then we fit into the circle ( A B C D ) straight lines equal to the straight line / L D / and placed continuously, there will be inscribed in the circle ( A B C D ) an equilateral polygon with an even number of sides which does not touch the lesser circle ( E F G H ). Q. E. F.

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