98.06 The relative size of the squares inscribed in a triangle

(1)

NOTES 117 With sin A

+

sin B

+

sin C

=

1:1 / Rr (circumradius R, inradius r), some

routine coordinate geometry leads eventually to the expected perpendicular distance r from the stationary point to any side. Furthermore, we have

elf elf

a

2f

ax2

=

ay2

>

0 and

ax ay

=

0

at I, so the Hessian, ~~ ~~ -

(a~2~yr

is positive here and I is indeed a minimum point.

Finally, note two sharp inequalities. For circumcentre and orthocentre we have

1

~ 4,

+

sinB sin C sin C sin A sin A sinB and

cos2A cos? B cos? C

1,

+

~

sinB sin C sin C sinA sinA sin B

respectively.

References

1. P. R. Allaire et al., Proving a nineteenth century ellipse identity, Math.

Gaz. 95 (March 2012) pp. 161-165.

2. 1. A. Scott, On the Lagrange method for constrained critical points,

Math. Gaz.

91

(March 2007) pp. 81-85.

J.

A. SeOTI

1 Shiptons Lane, Great Somerford, Chippenham SN15 5EJ

98.06 The relative size of the squares inscribed in a triangle

In this note, we consider a triangle

ABC

with area 1:1 and the square

::Ie

with two consecutive vertices on the line of

c,

a third vertex on

a,

and the last on b (Figure 1). The square erected outwardlh' on c is transformed into square

::I

eby the homothety of similarity ratio __ e - centered at C, where

he

+

C

he denotes the altitude perpendicular to c [1]. The side of ::Ieis thus

he

21:1

Se

=

--c

he+c c+he

Cambridge Core terms of use, available at https:/www.cambridge.org/core/terms. https://doi.org/10.1017/S0025557200000772 Downloaded from https:/www.cambridge.org/core. University of Basel Library, on 11 Jul 2017 at 09:13:21, subject to the

(2)

118 THE MATHEMATICAL GAZETTE

c

AGURE 1 AGURE2

We analyse the relative size of the squares ~a, ~b,and ~c' We prove in particular the result of [2] in a simpler, shorter, and more informative way: when the triangle is non-obtuse, the side lengths of the three squares differ by less than six percent from the longest square side.

One has for any triangle

~ =

b+hb

=

_----b+csi.na --f(b,c,a), _b_{> 0,}_c_{> 0, 0 <}_a

_<

_Jr.

Sb C

+

h; c

+

b sm

a

S· {)f c cos2

a

f' . I" fu . f

mce -;- = . 2' ISa strict y mcreasmg or constant ncnon 0

db (c

+

b sma)

b when c and a are fixed, with range (sin a,

suta)

for a T!' ~ and range {I}

for

a

= ~.

One has in particular Sb

=

Sc if, and only if, band c are the legs of an isosceles or right -angled triangle, with ~b = ~c in the latter case (in the right-angled comer).

We now consider a non-obtuse triangle, that is, c cos

a "

b "

cfsa

or

a

= ~.

We set

f(c

cos

a, c, a)

---cosa + sin

a

= g(a),

1 + cos

a

sin

a

which is invariant under

a ~

~ - a.

As

f

is an increasing function of b when c and a are fixed and as f(~, c, a) = lIf(b, c, a), the range of

f(b,

c, a)

=

sciSb for a non-obtuse triangle with fixed c and

a

is thus

[ ]., cosasina(sina-cosa) ..

g(a),l1 g(a). Since g (a)

=

.

2 ' g IS stnctly

(l+cosasma)

decreasing from 1 to

¥

on

[0, t]

before increasing symmetrically on

[i, t]·

Thus the ratio

sis'

of the sides

s " s'

of two squares inscribed in the same non-obtuse triangle is in the range from

¥

= 0.9428 to 1. The minimal value

¥

(for squares with areas in the ratio 8 : 9) corresponds to an isosceles right-angled triangle: the square of side S occupies then the middle third of the hypotenuse, whereas the side

s",

half as long as the legs, belongs to the square in the right-angled comer (Figure 2).

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NOTES 119

References

1. F. van Lamoen, Inscribed squares, Forum Geom., 4 (2004) pp. 207-214.

http://forumgeom.fau.eduIFG2004volume4IFG200424index.html

2. V. Oxman and M. Stupel, Why are the side lengths of the squares inscribed in a triangle so close to each other?, Forum Geom., 13 (2013) pp.113-1I5.

http://forumgeom.fau.eduIFG2013volume13IFG20131Iindex.html

GREGOIRE NICOLLIER

University of Applied Sciences of Western Switzerland, Route du Rawyl47, CH-1950 Sion, Switzerland

e-mail: [email protected]

98.07 The order of convergence of Newton's Method in

special cases

It is well known that in the general case the Newton-Raphson method for solving non-linear equations is of convergence order 2. (Actually Simpson in 1740 was the first to write the method in its modem form using a derivative, so perhaps it should be called the N-R-S method). Note that 'convergence order p' means that if an iteration method is of the form

Xi+ I

=

ljJ(xJ

and is converging to a root ~, then the order is defined aspif lim

!xi

+ 1 -

~I

= ljJ(P)(~)

i--?~

!xj -

~f

p!

(or equivalently if

lej

+11

=

k

lel

where

e,

=

x, -

o.

It follows that if

(1)

whereas

ljJ(P)(~) "" 0 then the order is indeed p.(For proof see e.g. [I].)

Now for Newton's method

f(x) ljJ(x)

=

x -

-f'

(x) and hence

CP' (~)

= 0 (unless!, (~) 0), while

CP" (~)

r (~)

_{f' (~) ""}

0 (usually).