G.161 circumscribe a circle about a square — Heath

IV.9

About a given square to circumscribe a circle.

Let < A B C D > be the given square; thus it is required to circumscribe a circle about the square < A B C D >.

For let / A C /, / B D / be joined, and let them cut one another at E.

Then, since / D A / is equal to / A B /, and / A C / is common, therefore the two sides / D A /, / A C / are equal to the two sides / B A /, / A C /; and the base / D C / is equal to the base / B C /;

therefore the angle < D A C ) is equal to the angle < B A C ). [G.29]

Therefore the angle < D A B ) is bisected by / A C /.

Similarly we can prove that each of the angles < A B C ), < B C D ), < C D A ) is bisected by the straight lines / A C /, / D B /.

Now, since the angle < D A B ) is equal to the angle < A B C ), and the angle < E A B ) is half the angle < D A B ), and the angle < E B A ) half the angle < A B C ), therefore the angle < E A B ) is also equal to the angle < E B A ); so that the side / E A / is also equal to / E B /. [G.27]

Similarly we can prove that each of the straight lines / E A /, / E B / is equal to each of the straight lines / E C /, / E D /.

Therefore the four straight lines / E A /, / E B /, / E C /, / E D / are equal to one another.

Therefore the circle described with centre E and distance one of the straight lines / E A /, / E B /, / E C /, / E D / will pass also through the remaining points; and it will have been circumscribed about the square < A B C D >.

Let it be circumscribed, as ( A B C D ).

Therefore about the given square a circle has been circumscribed.

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