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Mean Geometrically

A and B are two points on a circle centre O. Tangents at A and B cut at C. CO cuts the circle at D. What is the relationship between areas of ADBO, ABO and ACBO?

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So Big

One side of a triangle is divided into segments of length a and b by the inscribed circle, with radius r. Prove that the area is: abr(a+b)/ab-r^2

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Golden Triangle

Three triangles ABC, CBD and ABD (where D is a point on AC) are all isosceles. Find all the angles. Prove that the ratio of AB to BC is equal to the golden ratio.

Vecten

Stage: 5 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

This investigation is open ended and you can submit here anything at all that you find out. We are not expecting a complete solution from anyone. Indeed there are aspects and generalisations of this problem that no-one has yet explored fully.
vecten stage 1
There are lots of things to investigate. We hope you will find something interesting to contribute. Over the next few months we'll publish different bits and pieces and different methods until it adds up to a 'big picture' of this problem.

Squares are constructed on the sides of a triangle and the outer vertices are joined as shown in the diagram.

This is a problem that you might like to investigate using dynamic geometry software but whatever conjectures you are led to by the software will still need to be proved.

(1) Prove that all four triangles have the same area. (For convenience in referring to the diagram label the triangles $D, r, s$ and $t$ and angles $A, B, C, x, y$ and $z$ as shown).

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(2) Construct squares on the outer edges of the triangles and join the outer vertices of the squares to form three quadrilaterals. Find the angles in these quadrilaterals.

Try tessellating these quadrilaterals with copies of the four triangles, fitting triangles into the quadrilaterals like pieces of a jig-saw. What can you prove about these quadrilaterals and their areas?

(3) Construct another band of squares and quadrilaterals in the same way. Find the outer angles in the quadrilaterals in this band.

(4) If this construction is repeated indefinitely building bands of squares and quadrilaterals, ring upon ring spreading outwards, what else can you discover?

Thank you Geoff Faux for introducing this problem to us at the ATM teachers' conference last April where there was a lot of interest in it.
vecten stage 4