PROPOSITION I. PROBLEM.

In a given circle to draw a chord equal to a given straight line, which is not greater than the diameter of the circle.

Let ABC be the given O, and D the given line, not greater than the diameter of the O.

It is required to draw in the ABC a chord=D.

Draw EC, a diameter of o ABC.

Then if EC=D, what was required is done.

But if not, EC is greater than D. From EC cut off EF=D, and with centre E and radius EF describe a O AFB, cutting ABC in A and B ; and join AE.

the

Then, ·. E is the centre of ○ AFB,

.. EA=EF,

and .. EA=D.

Thus a chord EA equal to D has been drawn in © ABC.

Q. E. F.

Ex. Draw the diameter of a circle, which shall pass at a given distance from a given point.

PROPOSITION II. PROBLEM.

In a given circle to inscribe a triangle, equiangular to a given triangle.

G

H

Let ABC be the given O, and DEF the given ▲.

It is required to inscribe in ABC a ▲, equiangular to ▲ DEF.

Draw GAH touching the

ABC at the pt. A.

III. 17.

I. 23.

Make GAB= ▲ DFE, and ▲ HAC= 2 DEF.
Join BC. Then will ▲ ABC be the required A.

For

GAH is a tangent, and AB a chord of the O,

.. LACB= 4 GAB,

the

that is, 4 ACB= ▲ DFE.

So also, 4 ABC= ▲ HAC,

that is, ABC= ▲ DEF;

III. 32.

III. 32.

.. remaining ▲ BAC=remaining ▲ EDF ;

.. ▲ ABC is equiangular to ▲ DEF, and it is inscribed in

ABC.

Q. E. F.

Ex. If an equilateral triangle be inscribed in a circle, prove that the radii, drawn to the angular points, bisect the angles of the triangle.

PROPOSITION III. PROBLEM.

About a given circle to describe a triangle, equiangular to a given triangle.

Let ABC be the given O, and DEF the given ▲.

It is required to describe about the a▲ equiangular to ▲ EDF.

From O, the centre of the O, draw any radius OC.
Produce EF to the pts. G, H.

Make

L

I. 23.

COA = ▲ DEG, and ▲ COB= ▲ DFH.
Through A, B, C draw tangents to the O, meeting in L, M, N.
Then will LMN be the ▲ required.

For: ML, LN, NM are tangents to the O,
.. the s at A, B, C are rt. 4 S.

III. 18.

Now 4s of quadrilateral AOCM together=four rt. ≤ s.; and of these OAM and OCM are rt. 4S;

.. sum of 8 COA, AMC-two rt. 4 s.
But sum of 2 s ·DEG, DEF=two rt. ≤s;

.. sum of 4s COA, AMC=sum of 4 s DEG, DEF,
and COAL DEG, by construction;

.. LAMC ▲ DEF;

that is LMN= L DEF.

Similarly, it may be shewn that LNM= DFE; .. also MLN= LEDF.

I. 32.

Thus a▲, equiangular to ▲ DEF, is described about the .

Q. E. F.

Let ABC be the given ▲.

It is required to inscribe a ○ in the ▲ ABC.

Bisects ABC, ACB by the st. lines BO, CO, meeting

in 0.

From O draw OD, OE, OF, 1s to AB, BC, CA.
Then, in As EBO, DBO,

I. 9.

I. 12.

:: ▲ EBO= ▲ DBO, and ▲ BEO= ▲ BDO, and OB is common,

.. OE=OD.

Similarly it may be shewn that OE=OF.

I. 26.

If then a be described, with centre O, and radius OD, will pass through the pts. D, E, F ;

this

and the s at D, E and F are rt. 48, .. AB, BC, CA are tangents to the ;and thus a DEF may be inscribed in the ▲ ABC.

III. 16.

Q. E. F.

Ex. 1. Shew that, if OA be drawn, it will bisect the angle BAC.

Ex. 2. If a circle be inscribed in a right-angled triangle, the difference between the hypotenuse and the sum of the other sides is equal to the diameter of the circle.

Ex. 3. Shew that, in an equilateral triangle, the centre of the inscribed circle is equidistant from the three angular points. Ex. 4. Describe a circle, touching one side of a triangle and the other two produced. (NOTE. This is called an escribed circle.)

NOTE. Euclid's fifth Proposition of this Book has been already given on page 135.

PROPOSITION VI. PROBLEM.

To inscribe a square in a given circle.

Let ABCD be the given O.

It is required to inscribe a square in the ©.

Through O, the centre, draw the diameters AC, BD, 1 to each other.

Join AB, BC, CD, DA.

Then the 4s at O are all equal, being rt. 4 s,

.. the arcs AB, BC, CD, DA are all equal,

and.. the chords AB, BC, CD, DA are all equal; and ABC, being the in a semicircle, is a rt. 4.

I. Post. 4.

III. 26.

III. 29.

III. 31.

So also the 4s BCD, CDA, DAB are rt. ≤ s;

.. ABCD is a square,

and it is inscribed in the as was required.

Q. E. F.