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From the figure I observe that:
$i = i_1 + i_2 + x - 2$
$p = p_1 + p_2 - 2x + 2.$
Now, I shall calculate $F(P)$:
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(4) Now I shall prove the theorem for any triangle.
Let $T$ be a triangle. Any shape of triangle can be enclosed
in a rectangle with edges parallel to the coordinate axes, as in
the diagram, by adding at most three triangles.
From part (1): $$ F(T+T1+T2+T3) = F(T) + F(T1) + F(T2) +
F(T3). $$ But from part (2): $$ F(T+T1+T2+T3) = 0 $$ and from part
(3): $$ F(T1) = F(T2) = F(T3) = 0. $$ From these relations, I
observe that $F(T) = 0$.
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A and B are two fixed points on a circle and RS is a variable diamater. What is the locus of the intersection P of AR and BS?
Make a conjecture about the sum of the squares of the odd positive integers. Can you prove it?