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Flexi Quads

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

Flexi Quad Tan

As a quadrilateral Q is deformed (keeping the edge lengths constnt) the diagonals and the angle X between them change. Prove that the area of Q is proportional to tanX.

Air Routes

Find the distance of the shortest air route at an altitude of 6000 metres between London and Cape Town given the latitudes and longitudes. A simple application of scalar products of vectors.

Pythagoras on a Sphere

Age 16 to 18
Challenge Level

On a sphere of radius $R$ we use a scale factor and the equivalent formula is $$\cos {a\over R} = \cos {b\over R} \cos {c\over R}.$$
All the familiar trigonometric identities in Euclidean Plane Geometry have their counterparts in Spherical Geometry (otherwise known as Elliptic Geometry) and also in Hyperbolic Geometry. These are the geometries on flat surfaces (Euclidean Geometry), on surfaces of positive curvature (Spherical Geometry) and on surfaces of negative curvature (Hyperbolic Geometry). See the article Curvature of Surfaces.

Most of the results in these two other geometries are much more similar in form to the trigonometric identities you meet in school than the result proved here which is equivalent to Pythagoras' Theorem.

The corresponding Pythagorean Theorem for right-angled triangles in Hyperbolic Geometry is: $$\cosh a = \cosh b \cosh c.$$