Similarity and congruence

problem
Flower

Age

16 to 18

Challenge level

Six circles around a central circle make a flower. Watch the flower as you change the radii in this circle packing. Prove that with the given ratios of the radii the petals touch and fit perfectly.
problem
ArRh!

Age

14 to 16

Challenge level

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?
problem
Two triangles in a square

Age

14 to 16

Challenge level

Given that ABCD is a square, M is the mid point of AD and CP is perpendicular to MB with P on MB, prove DP = DC.
problem
A shade crossed

Age

14 to 16

Challenge level

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

Age

14 to 18

Challenge level

Given a regular pentagon, can you find the distance between two non-adjacent vertices?
problem
Chord

Age

16 to 18

Challenge level

Equal touching circles have centres on a line. From a point of this line on a circle, a tangent is drawn to the farthest circle. Find the lengths of chords where the line cuts the other circles.
problem
Overlap

Age

14 to 16

Challenge level

A red square and a blue square overlap. Is the area of the overlap always the same?
problem
Triangular tantaliser

Age

11 to 14

Challenge level

Draw all the possible distinct triangles on a 4 x 4 dotty grid. Convince me that you have all possible triangles.
problem
Parallel universe

Age

14 to 16

Challenge level

An equilateral triangle is constructed on BC. A line QD is drawn, where Q is the midpoint of AC. Prove that AB // QD.
problem
Matter of scale

Age

14 to 16

Challenge level

Can you prove Pythagoras' Theorem using enlargements and scale factors?