What happens to the perimeter of triangle ABC as the two smaller
circles change size and roll around inside the bigger circle?
Two semicircle sit on the diameter of a semicircle centre O of
twice their radius. Lines through O divide the perimeter into two
parts. What can you say about the lengths of these two parts?
A point P is selected anywhere inside an equilateral triangle. What
can you say about the sum of the perpendicular distances from P to
the sides of the triangle? Can you prove your conjecture?
Like Quad in
Quad (June 1998) the key to this
problem is using the fact that, when the midpoints of the sides of
a triangle are joined, the join is parallel to the third side and
half the length of the third side.
This solution came from Murat from
Due to the Thales relationship, SP and QR are both parallel to
BD, thus to each other. Moreover they are half as long as BD. A
similar relationship holds among PQ, RS and AC. Therefore PQRS is a
Let T be the midpoint of AC.
In the above figure, the areas of the triangles marked by the
same symbol are obviously equal (look for triangles on equal bases
which have the same height). Moreover Area (CST) =
S 1 , Area (CPT)= S
2 and Area (TPS)= S 3 . The
following equations are obviously true: