PROPOSITION II. PROBLEM.

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

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

It is required to inscribe in

to A DEF.

Draw GAH touching the

ABC a ▲, equiangular

ABC at the pt. A.

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

For GAH is a tangent, and AB a chord of the O,
.. LACB= L GAB,

III. 32.

that is, ACB= L DFE.

So also, ▲ ABC= ▲ HAC,

III. 32.

that is, ABC= ▲ DEF;

.. remaining BAC remaining 4 EDF;

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

M

B

A

H

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.

Make

Produce EF to the pts. G, H.

AOC= 2 DEG, and ▲ BOC= ¿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. 48.

Now 48 of quadrilateral AOCM together

=

III. 18. four rt. 48;

and of these LOAM and OCM are rt. 48;

.. sum of 8 AOC, AMC-two rt. 4 s. But sum of 48 DEG, DEF=two rt. 4s;

.. sum of 4s AOC, AMC=sum of 48 DEG, DEF,

I. 32.

:. LAMCL DEF;

that is, LMN= L DEF.

Similarly, it may be shewn that LNM= LDFE;

.. also MLN=LEDF.

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

Q. E. F.

It is required to inscribe a in the ▲ ABC. Bisect LS ABC, ACB by the st. lines BO, CO, meeting in O.

From O draw OD, OE, OF, 18 to AB, BC, CA.

Then, in As EBO, DBO,

:: LEBO= LDBO, and L BEO= L BDO, and OB is common,

this

.. 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;

and the 8 at D, E and F are rt. 48,

.. AB, BC, CA are tangents to the ;

III. 16.

and thus a DEF may be inscribed in the ▲ ABC.

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 15.

PROPOSITION VI. PROBLEM.

To inscribe a square in a given circle.

B

D

Let ABCD be the given O.

It is required to inscribe a square in the O.

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

Join AB, BC, CD, DA.

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

.. the arcs AB, BC, CD, DA are all equal, and.. the chords AB, BC, CD, DA are all equal;

III. 26.

III. 29.

and ABC, being the 4 in a semicircle, is a rt. 4.

III. 31.

So also the 48 BCD, CDA, DAB are rt. 48;

.. ABCD is a square,

and it is inscribed in the O as was required. Q. E. F.

PROPOSITION VII. PROBLEM.

To describe a square about a given circle.

Let ABCD be the given O, of which O is the centre. It is required to describe a square about the . Draw the diameters AC, BD, 1 to each other. Through A, B, C D draw EF, FG, GH, HE touching the O.

and.. AFB is a rt. 4.

So also the 4s at G, H and E are rt. 48.

I. 34.

Hence EFGH is a square, and it is described about the

Q. E. F.

Ex. In a given circle inscribe four circles, equal to each other, and in mutual contact with each other and with the given circle.

.