Let all the specified polygons be labeled clockwise.
Consider the capital letters as complex numbers and i = sqrt(-1).
We will make use of the fact that the product of a complex number
and plus i (minus i) is the complex number rotated CCW (CW) by a
right angle.
Q - A = (Q - K) + (K - C) + (C - A)
= (N - C) + (K - C) + (C - A)
= (A - C)(-i) + (B - C)(i) + (C - A)
= (C - A) + (B - A)(i)
= [(B - A) - (C - A)(i)](i)
= [(C - B)(-i) + (A - B)(i) + (B - A)](i)
= [(L - B) + (I - B) + (B - A)](i)
= [(P - I) + (I - B) + (B - A)](i)
= (P - A)(i)
Therefore, PAQ is an isosceles right triangle. Similar arguments
can be made for QBR and RCP.
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