Triangles

Triangle ABC is equilateral. D, the midpoint of BC, is the centre of the semi-circle whose radius is R which touches AB and AC, as well as a smaller circle with radius r which also touches AB and AC. What is the value of r/R?

Weekly Problem 13 - 2012
The diagram shows contains some equal lengths. Can you work out one of the angles?

Can you make a regular hexagon from yellow triangles the same size as a regular hexagon made from green triangles ?

Prove that, given any three parallel lines, an equilateral triangle always exists with one vertex on each of the three lines.

An irregular tetrahedron is composed of four different triangles. Can such a tetrahedron be constructed where the side lengths are 4, 5, 6, 7, 8 and 9 units of length?

Two right-angled triangles are connected together as part of a structure. An object is dropped from the top of the green triangle where does it pass the base of the blue triangle?

Weekly Problem 8 - 2010
Are you able to find triangles such that these five statements are true?

Find the sides of an equilateral triangle ABC where a trapezium BCPQ is drawn with BP=CQ=2 , PQ=1 and AP+AQ=sqrt7 . Note: there are 2 possible interpretations.

If I tell you two sides of a right-angled triangle, you can easily work out the third. But what if the angle between the two sides is not a right angle?