G.141 construct a circle segment on a chord admitting a given angle — Heath

III.33

On a given straight line to describe a segment of a circle admitting an angle equal to a given rectilineal angle.

Let / A B / be the given straight line, and the angle at < C ) the given rectilineal angle; thus it is required to describe on the given straight line / A B / a segment of a circle admitting an angle equal to the angle at < C ).

The angle at < C ) is then acute, or right, or obtuse.

First let it be acute, and, as in the first figure, on the straight line / A B /, and at the point A, let the angle < B A D ) be constructed equal to the angle at < C );

therefore the angle < B A D ) is also acute.

Let / A E / be drawn at right angles to / D A /, let / A B / be bisected at F, let / F G / be drawn from the point F at right angles to / A B /, and let / G B / be joined.

Then, since / A F / is equal to / F B /, and / F G / is common,

the two sides / A F /, / F G / are equal to the two sides / B F /, / F G /;

and the angle < A F G ) is equal to the angle < B F G );

[G.24]therefore the base / A G / is equal to the base / B G /.

Therefore the circle described with centre G and distance / G A / will pass through B also.

Let it be drawn, and let it be ( A B E ); let / E B / be joined.

Now, since / A D / is drawn from A, the extremity of the diameter / A E /, at right angles to / A E /,

[G.118]therefore / A D / touches the circle ( A B E ).

Since then a straight line / A D / touches the circle ( A B E ), and from the point of contact at A a straight line / A B / is drawn across in the circle ( A B E ), the angle < D A B ) is equal to the angle < A E B ) in the alternate segment of the circle. [G.140]

But the angle < D A B ) is equal to the angle at < C ); therefore the angle at < C ) is also equal to the angle < A E B ).

Therefore on the given straight line / A B / the segment ( A E B ) of a circle has been described admitting the angle < A E B ) equal to the given angle, the angle at < C ).

Next let the angle at < C ) be right;

and let it be again required to describe on / A B / a segment of a circle admitting an angle equal to the right angle at < C ).

Let the angle < B A D ) be constructed equal to the right angle at < C ), as is the case in the second figure;

let / A B / be bisected at F, and with centre F and distance either / F A / or / F B / let the circle ( A E B ) be described.

Therefore the straight line / A D / touches the circle ( A B E ), because the angle at < A ) is right. [G.118]

And the angle < B A D ) is equal to the angle in the segment ( A E B ), for the latter too is itself a right angle, being an angle in a semicircle. [G.139]

But the angle < B A D ) is also equal to the angle at < C ).

Therefore the angle < A E B ) is also equal to the angle at < C ).

Therefore again the segment ( A E B ) of a circle has been described on / A B / admitting an angle equal to the angle at < C ).

Next, let the angle at < C ) be obtuse;

and on the straight line / A B /, and at the point A, let the angle < B A D ) be constructed equal to it, as is the case in the third figure; let / A E / be drawn at right angles to / A D /, let / A B / be again bisected at F, let / F G / be drawn at right angles to / A B /, and let / G B / be joined.

Then, since / A F / is again equal to / F B /, and / F G / is common,

the two sides / A F /, / F G / are equal to the two sides / B F /, / F G /;

and the angle < A F G ) is equal to the angle < B F G );

[G.24]therefore the base / A G / is equal to the base / B G /.

Therefore the circle described with centre G and distance / G A / will pass through B also; let it so pass, as ( A E B ).

Now, since / A D / is drawn at right angles to the diameter / A E / from its extremity,

[G.118]/ A D / touches the circle ( A E B ).

And / A B / has been drawn across from the point of contact at A; therefore the angle < B A D ) is equal to the angle constructed in the alternate segment ( A H B ) of the circle. [G.140] But the angle < B A D ) is equal to the angle at < C ).

Therefore the angle in the segment ( A H B ) is also equal to the angle at < C ).

Therefore on the given straight line / A B / the segment ( A H B ) of a circle has been described admitting an angle equal to the angle at < C ).

Q. E. F.

Dependency Graph