7. The sum of the diameters of the inscribed and circumscribed circles of a right-angled triangle is equal to the sum of the sides containing the right angle.

8. If the circle inscribed in a triangle ABC touches the sides at D, E, F, shew that the triangle DEF is acute-angled; and express its angles in terms of the angles at A, B, C.

9. If is the centre of the circle inscribed in the triangle ABC, and the centre of the escribed circle which touches BC; shew that I, B, I, C are concyclic.

10. In any triangle the difference of two sides is equal to the difference of the segments into which the third side is divided at the point of contact of the inscribed circle.

11. In the triangle ABC the bisector of the angle BAC meets the base at D, and from the centre of the inscribed circle a perpendicular IE is drawn to BC: shew that the angle BID is equal to the angle CIE.

12. In the triangle ABC, I and S are the centres of the inscribed and circumscribed circles: shew that IS subtends at A an angle equal to half the difference of the angles at the base of the triangle.

13. In a triangle ABC, I and S are the centres of the inscribed and circumscribed circles, and AD is drawn perpendicular to BC: shew that Al is the bisector of the angle DAS.

14. Shew that the area of a triangle is equal to the rectangle contained by its semi-perimeter and the radius of the inscribed circle.

15. The diagonals of a quadrilateral ABCD intersect at O: shew that the centres of the circles circumscribed about the four triangles AOB, BOC, COD, DOA are at the angular points of a parallelogram.

16. In any triangle ABC, if | is the centre of the inscribed circle, and if Al is produced to meet the circumscribed circle at O; shew that O is the centre of the circle circumscribed about the triangle BIC.

17. Given the base, altitude, and the radius of the circumscribed circle; construct the triangle.

18. Describe a circle to intercept equal chords of given length on three given straight lines.

19. In an equilateral triangle the radii of the circumscribed and escribed circles are respectively double and treble of the radius of the inscribed circle.

20. Three circles whose centres are A, B, C touch one another externally two by two at D, E, F : shew that the inscribed circle of the triangle ABC is the circumscribed circle of the triangle DEF,

PROPOSITION 6. PROBLEM.

To inscribe a square in a given circle.

Bk

ΕΙ

Let ABCD be the given circle.

It is required to inscribe a square in the

Construction.

ABCD.

Find E the centre of the circle :

and draw two diameters AC, BD perp. to one another.

Join AB, BC, CD, DA.

Then the fig. ABCD shall be the square required.

Proof.

Because

For in the AS BEA, DEA,

BE = DE,

and EA is common;

III. 1.

I. 11.

I. Def. 15.

and the BEA = the DEA, being rt. angles;

Similarly it may be shewn that CD = DA, and that BC= CD. ... the fig. ABCD is equilateral.

And since BD is a diameter of the ABCD,
.. BAD is a semicircle;
... the BAD is a rt. angle.

III. 31.

Similarly the other angles of the fig. ABCD are rt. angles.

the fig. ABCD is a square;

and it is inscribed in the given circle.

[For Exercises see page 281.]

Q.E.F.

Let ABCD be the given circle.

It is required to circumscribe a square about the
Construction. Find E the centre of the

ABCD.

ABCD: III. 1.

and draw two diameters AC, BD perp. to one another. I. 11. Through A, B, C, D draw FG, GH, HK, KF perp. to EA, EB, EC, ED.

Then the fig. GK shall be the square required.

Proof. Because FG, GH, HK, KF are drawn perp. to radii at their extremities,

.. FG, GH, HK, KF are tangents to the circle.

And because the

AEB, EBG are both rt. angles,
GH is par1 to AC.

Similarly FK is par1 to AC :

and in like manner GF, BD, HK are par1.

III. 16.

Constr.

I. 28.

Hence the figs. GK, GC, AK, GD, BK, GE are parts.
... GF and HK each = BD;

also GH and FK each AC :

but AC = BD;

=

BEA is a rt. angle;

at G is a rt. angle.

Similarly the at F, K, H are rt. angles.

I. 34. Constr.

.. the fig. GK is a square, and it has been circumscribed

about the

ABCD.

Q.E.F.

Let ABCD be the given square.

It is required to inscribe a circle in the

Construction.

square ABCD.

Bisect the sides AB, AD at F and E.
Through E draw EH par1 to AB or DC:

1. 10.

I. 31.

and through F draw FK par1 to AD or BC, meeting EH at G.

Proof. Now ABAD, being the sides of a square;

Similarly it

and their halves are equal

Ax. 7.

Constr.

may be shewn that GE = GK, and GK = GH : .. GF, GE, GK, GH are all equal.

With centre G, and radius GE, describe a circle.
This circle must pass through the points F, E, K, H;
and it will be touched by BA, AD, DC, CB;
for GF, GE, GK, GH, being equal, are radii;
and the angles at F, E, K, H are rt. angles.
Hence the FEKH is inscribed in the sq. ABCD.

III. 16.

I. 29.

Q.E.F.

[For Exercises see p. 281.]

PROPOSITION 9. PROBLEM.

To circumscribe a circle about a given square.

Let ABCD be the given square.

It is required to circumscribe a circle about the square ABCD.

BAD.

that is, the diagonal AC bisects the

Similarly the remaining angles of the square are bisected

by the diagonals AC or BD.

Hence each of the

... the

EAD, EDA is half a rt. angle;

Similarly it

may

... EA, EB, EC, ED are all equal.

With centre E, and radius EA, describe a circle:

this circle must pass through the points A, B, C, D, and is therefore circumscribed about the sq. ABCD.

Q.E.F.