Similarity and congruence

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?

Find the missing distance in this diagram with two isosceles triangles

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

The diagonal of a square intersects the line joining one of the unused corners to the midpoint of the opposite side. What do you notice about the line segments produced?

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

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

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

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.

Weekly Problem 4 - 2008
In the figure given in the problem, calculate the length of an edge.

What fractions can you divide the diagonal of a square into by simple folding?