PROPOSITION XI.

To trisect the circumference of a circle.

G

R

Let GEA be the given circle.

It is required to trisect its circumference.

Join C the centre with any point A in the circumference.

With A as centre at distance AC describe a circle, cutting GEA in P and Q.

Produce AC to meet GEA in the point R.

Then shall the circumference of GEA be trisected in P, Q, R.

Join CP, CQ, AP, AQ.

Then As ACP, ACQ are equilateral, and .. equiangular ;

(I. 5)

.. each of the Ls ACP, ACQ is = a third of two

.. Ls PCR, QCR, PCQ are equal to one another. .. the arcs PR, QR, PQ are equal to one another.

(III. 8) Wherefore the circumference GEA has been trisected.

Q.E.F.

COROLLARY.-Hence the circumference of a circle may be

divided into 6, 12, 24, &c., equal parts.

If PC, QC be produced to meet the given circumference GEA,

it will be divided into six equal arcs.

And so on.

BOOK IV.

INSCRIBED AND CIRCUMSCRIBED FIGURES.

DEFINITION.

A circle is said to be described about a rectilineal figure when the circumference of the circle passes through all the angular points of the figure.

PROPOSITION I.

To describe a circle about a given triangle.

R

Let ABC be the given triangle. It is required to describe a circle about the ▲ ABC.

Bisect AC in Q and AB in R.

(I. 9)

(I. 12)

Through Q and R draw straight lines perpendicular to AC and AB intersecting in O.

Join OA, OB, OC.

With O as centre and OA as radius describe a circle ; it shall be the one required.

Because AR, RO, and L ARO are respectively equal to BR, RO, and ▲ BRO,

.. OA = ОВ.

(I. 4)

Similarly OA = OC;

.. the circle passes through B and C, and .. is de

scribed about the ▲ ABC.

Wherefore a circle &c.

Q.E.F.

DEFINITION.

A circle is said to be inscribed in a rectilineal figure when each side of the figure touches the circle.

PROPOSITION II.

To inscribe a circle in a given triangle.

F

D

R

Let PQR be the given triangle; it is required to inscribe a circle within it.

C.

Bisects PQR, PRQ by QC, RC, intersecting in

(I. 8)

From C let fall CD, CE, CF perpendicular to QR, RP, PQ.

(I. 13)

With C as centre and CD as radius describe a circle; it shall be the one required.