G.246 equal circles angles — Heath

VI.33

In equal circles angles have the same ratio as the circumferences on which they stand, whether they stand at the centres or at the circumferences.

Let ( A B C ), ( D E F ) be equal circles, and let the angles < B G C ), < E H F ) be angles at their centres G, H, and the angles < B A C ), < E D F ) angles at the circumferences;

I say that, as the circumference ( B C ) is to the circumference ( E F ), so is the angle < B G C ) to the angle < E H F ), and the angle < B A C ) to the angle < E D F ).

For let any number of consecutive circumferences ( C K ), ( K L ) be made equal to the circumference ( B C ), and any number of consecutive circumferences ( F M ), ( M N ) equal to the circumference ( E F ); and let / G K /, / G L /, / H M /, / H N / be joined.

Then, since the circumferences ( B C ), ( C K ), ( K L ) are equal to one another,

[G.134]the angles < B G C ), < C G K ), < K G L ) are also equal to one another;

therefore, whatever multiple the circumference ( B L ) is of ( B C ), that multiple also is the angle < B G L ) of the angle < B G C ).

For the same reason also,

whatever multiple the circumference ( N E ) is of ( E F ), that multiple also is the angle < N H E ) of the angle < E H F ).

If then the circumference ( B L ) is equal to the circumference ( E N ), the angle < B G L ) is also equal to the angle < E H N ); [G.134] if the circumference ( B L ) is greater than the circumference ( E N ), the angle < B G L ) is also greater than the angle < E H N ); and, if less, less.

There being then four magnitudes, two circumferences ( B C ), ( E F ), and two angles < B G C ), < E H F ), there have been taken, of the circumference ( B C ) and the angle < B G C ) equimultiples, namely the circumference ( B L ) and the angle < B G L ), and of the circumference ( E F ) and the angle < E H F ) equimultiples, namely the circumference ( E N ) and the angle < E H N ).

And it has been proved that, if the circumference ( B L ) is in excess of the circumference ( E N ), the angle < B G L ) is also in excess of the angle < E H N ); if equal, equal; and if less, less.

Therefore, as the circumference ( B C ) is to ( E F ), so is the angle < B G C ) to the angle < E H F ). [G.176]

But, as the angle < B G C ) is to the angle < E H F ), so is the angle < B A C ) to the angle < E D F ); for they are doubles respectively.

Therefore also, as the circumference ( B C ) is to the circumference ( E F ), so is the angle < B G C ) to the angle < E H F ), and the angle < B A C ) to the angle < E D F ).

Therefore etc. Q. E. D.

Dependency Graph

Required for

G.586, G.587, G.588, G.589, G.597, G.601