Similarity and congruence

The largest square which fits into a circle is ABCD and EFGH is a square with G and H on the line CD and E and F on the circumference of the circle. Show that AB = 5EF. Similarly the largest equilateral triangle which fits into a circle is LMN and PQR is an equilateral triangle with P and Q on the line LM and R on the circumference of the circle. Show that LM = 3PQ

Try out this geometry problem involving trigonometry and number theory

Can you spot a cunning way to work out the missing length?

Explain how the thirteen pieces making up the regular hexagon shown in the diagram can be re-assembled to form three smaller regular hexagons congruent to each other.

Find the area of the shaded region created by the two overlapping triangles in terms of a and b?

A new problem posed by Lyndon Baker who has devised many NRICH problems over the years.

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.

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

If the sides of the triangle in the diagram are 3, 4 and 5, what is the area of the shaded square?