Areas between Two Parallelograms

What Might This Be About?

4 February 2016, Created with GeoGebra

Problem

A smaller parallelogram $OA'B'C'\;$ is inside a larger parallelogram $OABC\;$ such that they share a common vertex $O\;$ along with two of their adjacent edges as shown in the figure. $A'B'\;$ and $C'B'\;$ are produced to meet the edges of the outer parallelogram at $P\;$ and $Q\;$ respectively.

Areas between Two Parallelograms  - problem

Show that the area of the triangle $OPQ\;$ is half that of the $L-shaped\;$ region between the two parallelograms.

Solution

Let $[F]\;$ denote the area of shape $F.\;$ Then

Areas between Two Parallelograms  - solution

By Euclid I.37,

$\begin{align} [\Delta B'OP]&=[\Delta B'O_1P]=[\Delta B'C'P]=\frac{1}{2}[C'B'PC] \\ \; [\Delta B'OQ]&=[\Delta B'O_2Q]=[\Delta B'A'Q]=\frac{1}{2}[A'AQB'] \\ \; [\Delta PB'Q]&=\frac{1}{2}[B'QBP], \end{align}$

But then

$\begin{align} [\Delta POQ]&=[\Delta B'OP]+[\Delta B'OQ]+[\Delta PB'Q]\\ &=[\Delta B'C'P]+[\Delta B'A'Q]+[\Delta PB'Q]\\ &=\frac{1}{2}([C'B'PC]+[A'AQB']+[B'QBP]). \end{align}$

Acknowledgment

The statement has been communicated to me by G. S. Ranganath, with the solution by his son G. R. Jayanth.

Extra

Here's a related problem from an upcoming book Population Explosion and Other Mathematical Puzzles (World Scientific, 2016) by Dick Hess.

The figure below show a lake $OA'AQPCC'\;$ That is nearly a parallelogram except for a segment of its shore line. Let $S_1=[CC'QP],\;$ $S_2=[A'AQP],\;$ and $S_3=[OPQ].\;$ Express $S_3\;$ in terms of $S_1\;$ and $S_2.$

Areas between Two Parallelograms  - extra

Answer: Areas between Two Parallelograms  - answer to extra

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