Start with any triangle T1 and its inscribed circle. Draw the
triangle T2 which has its vertices at the points of contact between
the triangle T1 and its incircle. Now keep repeating this process
starting with T2 to form a sequence of nested triangles and
circles. What happens to the triangles? You may like to investigate
this interactively on the computer or by drawing with ruler and
compasses. If the angles in the first triangle are a, b and c prove
that the angles in the second triangle are given (in degrees) by
f(x) = (90 - x/2) where x takes the values a, b and c. Choose some
triangles, investigate this iteration numerically and try to give
reasons for what happens. Investigate what happens if you reverse
this process (triangle to circumcircle to triangle...)
M is any point on the line AB. Squares of side length AM and MB are
constructed and their circumcircles intersect at P (and M). Prove
that the lines AD and BE produced pass through P.
A kite shaped lawn consists of an equilateral triangle ABC of side 130 feet and an isosceles triangle BCD in which BD and CD are of length 169 feet. A gardener has a motor mower which cuts strips of grass exactly one foot wide and wishes to cut the entire lawn in parallel strips. What is the minimum number of strips the gardener must mow?
The largest square which fits into a circle is $ABCD$ and $EFGH$
is a square with $E$ and $F$ on the line $AB$ and $G$ and $H$ 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 $LM = 3PQ$.