Similarity and congruence

problem
Triangle midpoints

Age

14 to 16

Challenge level

You are only given the three midpoints of the sides of a triangle. How can you construct the original triangle?
problem
Look before you leap

Age

14 to 16

Challenge level

Can you spot a cunning way to work out the missing length?
problem
Double angle triples

Age

16 to 18

Challenge level

Try out this geometry problem involving trigonometry and number theory
problem
Strange rectangle

Age

16 to 18

Challenge level

ABCD is a rectangle and P, Q, R and S are moveable points on the edges dividing the edges in certain ratios. Strangely PQRS is always a cyclic quadrilateral and you can find the angles.
problem
Fitting in

Age

14 to 16

Challenge level

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
problem
Golden triangle

Age

16 to 18

Challenge level

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.
problem
Two trees

Age

16 to 18

Challenge level

Two trees 20 metres and 30 metres long, lean across a passageway between two vertical walls. They cross at a point 8 metres above the ground. What is the distance between the foot of the trees?
problem
Is a square a rectangle?

Age

7 to 11

Challenge level

How many rectangles can you find in this shape? Which ones are differently sized and which are 'similar'?
problem
Doubly isosceles

Age

14 to 16

Challenge level

Find the missing distance in this diagram with two isosceles triangles
problem
Triangular slope

Age

14 to 16

Challenge level

Can you find the gradients of the lines that form a triangle?