G.158 inscribe a square in a circle — Heath

IV.6

In a given circle to inscribe a square.

Let ( A B C D ) be the given circle; thus it is required to inscribe a square in the circle ( A B C D ).

Let two diameters / A C /, / B D / of the circle ( A B C D ) be drawn at right angles to one another, and let / A B /, / B C /, / C D /, / D A / be joined.

Then, since / B E / is equal to / E D /, for E is the centre, and / E A / is common and at right angles, therefore the base / A B / is equal to the base / A D /. [G.24]

For the same reason each of the straight lines / B C /, / C D / is also equal to each of the straight lines / A B /, / A D /;

therefore the quadrilateral < A B C D > is equilateral.

I say next that it is also right-angled.

For, since the straight line / B D / is a diameter of the circle ( A B C D ), therefore ( B A D ) is a semicircle;

[G.139]therefore the angle < B A D ) is right.

For the same reason each of the angles < A B C ), < B C D ), < C D A ) is also right;

therefore the quadrilateral < A B C D > is right-angled.

But it was also proved equilateral; therefore it is a square; [G.63] and it has been inscribed in the circle ( A B C D ).

Therefore in the given circle the square < A B C D > has been inscribed.

Q. E. F.

Dependency Graph

Required for

G.563, G.565, G.570, G.571, G.572, G.573