Triangles

The centre of the larger circle is at the midpoint of one side of an equilateral triangle and the circle touches the other two sides of the triangle. A smaller circle touches the larger circle and two sides of the triangle. If the small circle has radius 1 unit find the radius of the larger circle.

If you know the sizes of the angles marked with coloured dots in this diagram which angles can you find by calculation?

From the measurements and the clue given find the area of the square that is not covered by the triangle and the circle.

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

Triangle ABC is an equilateral triangle with three parallel lines going through the vertices. Calculate the length of the sides of the triangle if the perpendicular distances between the parallel lines are 1 unit and 2 units.

A floor is covered by a tessellation of equilateral triangles, each having three equal arcs inside it. What proportion of the area of the tessellation is shaded?

ABCDEFGH is a 3 by 3 by 3 cube. Point P is 1/3 along AB (that is AP : PB = 1 : 2), point Q is 1/3 along GH and point R is 1/3 along ED. What is the area of the triangle PQR?

Prove that a triangle with sides of length 5, 5 and 6 has the same area as a triangle with sides of length 5, 5 and 8. Find other pairs of non-congruent isosceles triangles which have equal areas.