11 Geometry

I will not add much on Geometry, the oldest of the formal mathematics disciplines. It is still the strongest of the IMO disciplines, where normally two of the six problems are generally geometric, with Euclidean proof normally required (although if a coordinate proof is complete it can attract 7 rather than otherwise 0 points). I do not recall three dimensional Euclidean geometry since the mid 90s, but will discuss this in Chapter 12 a little.

However Geometry has suffered massive cuts in the syllabi of various countries since the 1980s. This has happened to the extent that the AMC has not been able to set circle geometry problems for years 9 and 10 because it ceased to be taught even in strongly developed states like New South Wales.

Teachers who supported Geometry's downgrade said the reason was simple, people do not use geometry in later life. This overlooks the fact that Geometry, with its theorems, logic and structure were the main branch of mathematics for developing logical reasoning, a vital skill in later life.

I think Geometry has also suffered because the textbooks have been very dry. Old fashioned, theorem, proof, theorem proof, exercise, etc but no motivating discussion. It doesn't have to be like that. The following famous AMC problem from the 1980s, set by Bob Bryce, has already been mentioned in Chapter 3 of the history part of this site and shows that the circle geometry has a role in solving real world problems. It was well known that the Southern Cross can be seen in some northern latitudes as French Aviation pioneers used it for navigation when crossing the Sahara in establishing routes to South America. But how far north can it be seen?

Example 11.1

The latitude of Canberra is 35^o19'S. At its highest point in the sky when viewed from Canberra the lowest star in the Southern Cross is 62^o20' above the southern horizon. It can be assumed that rays of light from this star to any point on the earth are parallel. The northernmost latitude at which the complete Southern Cross can be seen is

(A) 0^o (B) 27^o01'S (C) 27^o01'N (D) 7^o39'N (E) 7^o39'S

Solution 11.1

The northernmost point is where a ray of light from the star is tangential to the earth's surface, i.e. point A in the diagram. The object is to determine the value of x in the diagram.

Consider the quadrilateral OABC in the diagram. Because the angles at A and C are 90^o, the remaining two angles must add to 180^o. Therefore x^o+35^o19'+∠ABC=180^o.

Because the rays reaching points A and C are parallel it is clear that ∠ABC=180^o-62^o20'. Therefore x^o+35^o19'+180^o-62^o20'=180^o, giving x^o=62^o20'-35^o19'=27^o01', Hence (C).

Our Comments from the Time

We have been assured that students familiar with French literature were readily able to reduce the options to (C) and (D), i.e. the two answers with northern latitudes. The well-known French novel Vol de Nuit (Night Flight), by Antoine de Saint-Exupéry tells the story of the adventure of the first intercontinental link between Europe and America (Paris--Dakar(Senegal)--Natal(Brazil)). The author was a pilot who pioneered the link. There are frequent references in the book to the use of the Southern Cross in navigation over the Sahara, which is in the northern hemisphere.