Proposition IV.6 — In a given circle to inscribe a square.

Proof

Let

A B C D

be the given circle with centre

E

. Draw two diameters

A C

and

B D

at right angles (I.11). Join

A B

B C

C D

D A

. The four right triangles at

E

are congruent by I.4 (two radii and the common right angle), so

A B = B C = C D = D A

. The inscribed angles standing on the diameters are right (III.31), so all four angles of

A B C D

are right. Therefore

A B C D

is a square.

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Depends on (4)

I.4Proposition I.4If two triangles have two sides equal to two sides respectively, and have the angles contained by the equal straight…
I.11Proposition I.11To draw a straight line at right angles to a given straight line from a given point on it.
III.31Proposition III.31In a circle the angle in the semicircle is right, that in a greater segment less than a right angle, and that in a less…
I.15Definition I.15A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among…

Required by (dependents) (1)

XIII.15Proposition XIII.15To construct a cube and comprehend it in a sphere, as in the preceding case; and to prove that the square on the…

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