During one of my recent mathematical explorations, I wanted to calculate the distance of the points of an isosceles triangle to its orthocentre (points where the altitudes meet).

ðŸ™‚

In other words, I firstly wanted to find in terms of in the following figure.

Once is found, the required distances will be and . In this post I will show two ways of finding .

Here is the approach I initially came up with. Since the angles at and are each degrees, quadrilateral is cyclic. Then by the intersecting chords theorem (or by the similarity of triangles and ),

from which Â Â Â Â Â Â …(1)

Also, since triangles and are similar,

from which Â Â Â Â Â Â …(2)

Combining (1) and (2),

.

It follows that . Who would have thought that such a simple answer would be found! Hence, we have rediscovered a way of constructing the reciprocal of a positive number : draw an isosceles triangle with base length and height , then the height of its orthocentre above the base is .

Naturally, the question comes up of whether there is an easier way of calculating this answer and surely enough there is. If we extend until it meets the circumcircle of at , we find that are equal (each being the complement of ). Then we also have , both angles lying on the arc . Both acute and obtuse angled cases are illustrated below.

It follows by Angle-Angle-Side that triangles HPB and C’PB are congruent, from which . In other words, the reflection of in lies on the circumcircle! This fact is true for the reflection of in any side, and need not be isosceles.

By the intersecting chords theorem for and intersecting at (or equivalently, by the similarity of triangles and ),

or as required.

I later found that the second proof is also given in pp18-19 of Honsberger’s book entitled “Episodes in 19th and 20th century Euclidean Geometry”.

After watching the movie Fire in Babylon (I have now seen it twice – well worth seeing if you are a cricket fan!) I was curious to find out the country of origin of prominent players of the West Indies. There are some 15 countries that make up the West Indies (English speaking part of the Caribbean, note that it does not include Bermuda, Bahamas, Turks and Caicos Islands or the Cayman Islands). Some of which are dependent territories – these are shown in italics below. The six first class teams of the West Indies are:

Leeward Islands: Antigua and Barbuda, St Kitts and Nevis, Anguilla, Montserrat, British Virgin Islands, US Virgin Islands, Sint Maarten

Windward Islands: Dominica, Grenada, St Lucia, St Vincent and the Grenadines

Of these nations, the only ones never to have a test cricket representative are the British and US Virgin Islands and Sint Maarten. See here for a full list of West Indian test players by origin.

From the table see how many great players the tiny nations of Barbados and Antigua and Barbuda produced!

Given a point inside an angle, it easy to come up with many minimisation problems to solve. Here are some mostly taken from [1], with the answers given below, but proofs are left as exercises to the interested reader. ðŸ™‚

Let be the angle, the point inside the angle and be a variable straight line through with on and on . Let be variable points on respectively.

Problems:

Find so that the area of triangle is minimised.

Find so that the perimeter of triangle is minimised.

Find so that the length is minimised.

Find so that is minimised.

Find so that is maximised.

Find so that is minimised (p, q > 0).

Find so that and is minimised.

Find so that the perimeter of triangle is minimised.

Do try at least one of these before reading below for the answers!

Answers without proof:

[to minimise the area of ] Choose so that (extend to twice its length, then complete the parallelogram with as centre and as diagonal).

[to minimise the perimeter of ] Construct a circle tangent to the angle and through , then is its tangent through . The circle is formed by first choosing any point on the angle bisector and drawing any initial circle tangent to (its radius can be found by dropping a perpendicular to ). This initial circle can be scaled up or down by drawing parallel lines so that the final circle passes through .

[to minimise the length of ] The solution is Philo’s line, mentioned in a previous blog entry. The points and cannot be found by straight edge and compass, but should be chosen so that they are equidistant to the midpoint of .

[to minimise ] Choose so that is isosceles (construct the angle bisector of angle , then draw the perpendicular to this line through ).

[to maximise ] Choose perpendicular to .

[to minimise ] Choose so that . Given the lengths drawn from this can be done via parallel lines and similar triangles as shown:

[to minimise so that ] Rotate by about to . Then is found by the intersection of and and is constructed so that .

[to minimise the perimeter of triangle ] Reflect in and . Then and are where the line joining these reflected images meets and . If then .

So given the same setup we have at least 8 different problems with 8 different constructions and answers!

Reference

[1] T. Andreescu, O. Mushkarov, L. Stoyanov, Geometric Problems on Maxima and Minima, BirkhÃ¤user, 2006.

Firstly, I thought I would see what this year’s AFL ladder would look like if each team played each other once (in reality with 17 teams playing 22 games each, a team plays six others twice and the other ten once). To do this I simply halved the results (scores and premiership points) of those games between teams that played each other twice during the home and away season. Here is that “normalised” ladder in the final two columns compared with the actual one. It is as though each team played every other once, so that the maximum possible number of points is 16*4 = 64.

Â #

TEAM

Played

Wins

Losses

Draws

Â %

PTS

Norm Pts

Norm %

1

Collingwood

22

20

2

0

167.66

80

60

168.82

2

Geelong

22

19

3

0

157.38

76

54

158.85

3

Hawthorn

22

18

4

0

144.12

72

52

139.46

4

West Coast

22

17

5

0

130.32

68

46

121.83

5

Carlton

22

14

7

1

130.88

58

41

130.13

6

St Kilda

22

12

9

1

112.76

50

38

110.94

7

Sydney

22

12

9

1

109.34

50

36

110.12

8

Essendon

22

11

10

1

100

46

35

103.02

9

North Melbourne

22

10

12

0

101.15

40

30

103.94

10

Western Bulldogs

22

9

13

0

95.59

36

30

101.13

11

Fremantle

22

9

13

0

83.11

36

28

85.66

12

Richmond

22

8

12

1

86.35

34

22

83.84

13

Melbourne

22

8

13

1

85.27

34

26

84.84

14

Adelaide

22

7

15

0

79.43

28

18

77.97

15

Brisbane Lions

22

4

18

0

80.98

16

10

78.30

16

Port Adelaide

22

3

19

0

64.51

12

8

65.41

17

Gold Coast

22

3

19

0

56.27

12

10

56.25

We see that the top 8 would be unchanged, and only Richmond-Melbourne and Gold Coast-Port Adelaide would have been swapped. This year, no team can really complain about having an unfortunate draw.

Secondly, after reading that Geelong has compiled a 105-20 (84%) win-loss record in their past 5 years (in fact winning 103 of their last 120 games including streaks of 15, 15, 13, 13, 12!), I thought I would compare it with a few other top team sports results over a five year period. We are not necessarily looking at teams that won championships in every year of that period, more winning percentage. Games with many draws such as test cricket, ice hockey and soccer had to be excluded since they give lower winning percentages. Note that a draw or no result counts as half a win in the percentages below. It can be seen that Geelong’s winning percentage compares favourably with those of many great sporting teams of the past.

The New England patriots compiled a 77-17 (81.9%) record from 2003-2007 [ref].

The Australian one-day cricket team had a 107-26 win-loss record plus 2 ties and 5 no results (78.9%) between 2001 and 2005 [ref].

Between 1981 and 1986 the Boston Celtics had a 363-128 record (73.9%) including playoff games. As far as I can tell this exceeds any five-year period during their dynasty of the 1950s and 60s, as well as any best 5-year period of the LA Lakers or Chicago Bulls.

The St George Dragons had a 87-1-13 win-draw-loss record (86.6%) in Australian Rugby League between 1957 and 1961 (they won 11 consecutive premierships from 1956 to 1966!)

Worth a mention, tennis doubles “team” Martina Navratilova and Pam Shriver had a mind-boggling 217-5 (97.7%) record from 1983 to 1987. This included 109 consecutive wins between April 1983 and Wimbledon 1985 in which they only lost 14 sets! (Data taken manually from the WTA website, do correct me if I am wrong!)