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 quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4.
Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot be greater than sqrt2.
Find a quadrilateral of this type for which s2 is approximately sqrt3and show that s2 is always less than sqrt3.
Find a quadrilateral of this type for which s3 is approximately 2 and show that s2 is always less than 2.
Find a quadrilateral of this type for which s4=2 and show that s4 cannot be greater than 2.
Tony Cardell, has sent in this solution,
which gives the correct answer. We're still not quite convinced:
how does he know
that the strips are parallel to the longer
side? If anyone can explain this, we'll add their explanation
We must find the maximum distance between a pair of opposite
sides (it doesn't matter which since this is a kite). If we extend
the two sides (coloured blue), the intersection is on the same side
of the kite as the equilateral triangle, as shown. So the maximum
distance between the two sides is the red line.
Now look at the isosceles triangle part of the kite. One formula
for the area of a triangle with sides a,b,c says that if s is the
semi-perimeter, (a+b+c)/2, then the area is the square root of
s(s-a)(s-b)(s-c). The isosceles triangle has sides 169, 169, 130,
so s=234 and the area is 10140 square feet. But we also know that
the area is 1/2 x base x height, so the height we want is 10140 x 2
/ 169=120. So 120 foot-wide strips will be needed.