:

B

Let ABC be the triangle, in which a circle is to be inscribed.

Construction. Bisect the ABC, ACB by the st. lines Bl Cl, which intersect at I.

Then I is the centre of the required circle.

Proof. From I draw ID, IE, IF perp. to BC, CA, AB.

Prob. 1.

Then every point in BI is equidistant from BC, BA; Prob. 15.

... ID=IF.

And every point in Cl is equidistant from CB, CA;
.'. ID=IE.

... ID, IE, !F are all equal.

With centre I and radius ID draw a circle ; this will pass through the points E and F. Also the circle will touch the sides BC, CA, AB, because the angles at D, E, F are right angles.

.. the DEF is inscribed in the ABC.

NOTE. From II., p. 96 it is seen that if Al is joined, then Al bisects the angle BAC: hence it follows that

The bisectors of the angles of a triangle are concurrent, the point of intersection being the centre of the inscribed circle.

DEFINITION.

A circle which touches one side of a triangle and the other two sides produced is called an escribed circle of the triangle.

Let ABC be the given triangle of which the sides AB, AC are produced to D and E.

It is required to describe a circle touching BC, and AB, AC produced.

Construction. Bisect the CBD, BCE by the st. lines Bl1, Cl, which intersect at l

Then I is the centre of the required circle.

Proof. From 1 draw IF, IG, IH perp. to AD, BC, AE. Then every point in Bl, is equidistant from BD, BC; Prob. 15. ... IF = IG.

Similarly GI2H.

=

.. IF, IG, IH are all equal.

With centre, and radius IF describe a circle;
this will pass through the points G and H.
Also the circle will touch AD, BC, and AE,
because the angles at F, G, H are rt. angles.

.. the OFGH is an escribed circle of the ▲ ABC.

NOTE 1. It is clear that every triangle has three escribed circles. Their centres are known as the Ex-centres.

NOTE 2. It may be shewn, as in II., page 96, that if Al, is joined, then Al, bisects the angle BAC: hence it follows that

The bisectors of two exterior angles of a triangle and the bisector of the third angle are concurrent, the point of intersection being the centre of an escribed circle.

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

B

Let ABC be the given circle, and DEF the given triangle.

=

D.

Analysis. A ▲ ABC, equiangular to the A DEF, is inscribed in the circle, if from any point A on the Oe two chords AB, AC can be so placed that, on joining BC, the B = the LE, and the C the F; for then the A= the Theor. 16. Now the B, in the segment ABC, suggests the equal angle between the chord AC and the tangent at its extremity (Theor. 49.); so that, if at A we draw the tangent GAH,

Reversing these steps, we have the following construction.

Construction. At any point A on the Oce of the ABC draw the tangent GAH.

Prob. 22.

NOTE. In drawing the figure on a larger scale the student should shew the construction lines for the tangent GAH and for the angles GAB, HAC. A similar remark applies to the next Problem.

PROBLEM 29.

About a given circle to circumscribe a triangle equiangular to a given triangle.

Let ABC be the given circle, and DEF the given triangle.

Analysis. Suppose LMN to be a circumscribed triangle in which the M-the E, the N = the LF, and consequently, the L= the D.

Let us consider the radii KA, KB, KC, drawn to the points of contact of the sides; for the tangents LM, MN, NL could be drawn if we knew the relative positions of KA, KB, KC, that is, if we knew the ▲ BKA, BKC.

Now from the quad' BKAM, since the ▲ B and A are rt. 4o,

Construction.

Produce EF both ways to G and H.

Find K the centre of the O ABC,

and draw any radius KB.

At K make the 4 BKA equal to the

DEG;

and make the BKC equal to the DFH.

Through A, B, C draw LM, MN, NL perp. to KA, KB, KC.
Then LMN is the required triangle.

[The student should now arrange the proof synthetically.]

EXERCISES.

ON CIRCLES AND TRIANGLES.

(Inscriptions and Circumscriptions.)

1. In a circle of radius 5 cm. inscribe an equilateral triangle; and about the same circle circumscribe a second equilateral triangle. In each case state and justify your construction.

2. Draw an equilateral triangle on a side of 8 cm., and find by calculation and measurement (to the nearest millimetre) the radii of the inscribed, circumscribed, and escribed circles.

Explain why the second and third radii are respectively double and treble of the first.

3. Draw triangles from the following data:

(i) a=2.5′′, B=66°, C=50°;

(ii) a=2.5", B=72°, C=44°;

(iii) a=2.5′′, B=41°, C=23°.

Circumscribe a circle about each triangle, and measure the radii to the nearest hundredth of an inch. Account for the three results being the same, by comparing the vertical angles.

4. In a circle of radius 4 cm. inscribe an equilateral triangle. Calculate the length of its side to the nearest millimetre; and verify by measurement.

Find the area of the inscribed equilateral triangle, and shew that it is one quarter of the circumscribed equilateral triangle.

5. In the triangle ABC, if I is the centre, and r the length of the radius of the in-circle, shew that

AIBC=ar; AICA=}br;` AIAB=cr.

Verify this formula by measurements for a triangle whose sides are 9 cm., 8 cm., and 7 cm.

6. If r1 is the radius of the ex-circle opposite to A, prove that

If a=5 cm.,

ABC= (b+c-a) r1.

b=4 cm., c=3 cm., verify this result by measurement.

7. Find by measurement the circum-radius of the triangle ABC in which a 63 cm., b=30 cm., and c=5.1 cm.

Draw and measure the perpendiculars from A, B, C to the opposite sides. If their lengths are represented by P1, P2, P3, verify the following