Happiness and Unhappiness

A woodcutter went up the hills to cut trees for his living everyday. One day, he went to the hills as usual. On his way, he saw a wounded little bird. The little bird's body was covered with glittering silver feathers. It was very beautiful. The woodcutter said, " I've never seen such a beautiful bird before." So he took the little bird home and cured its wounds. During the rehabilitation process, the little bird sang to the woodcutter sweet and beautiful songs. The woodcutter felt very very happy.
One day, the woodcutter's neighbour came to visit him and saw the silver bird. He said to the woodcutter, " I've seen a little bird with golden feathers. That gold bird was much more beautiful than your silver bird. Besides, it sang much better than yours."
The woodcutter wondered, "Is there really such a gold bird? How beautiful is this gold bird? Can it sing very beautiful songs?" From then on, the woodcutter began not to enjoy songs sang by the silver bird. Gradually, the woodcutter became more and more unhappy each day.
Now the silver bird fully recovered from its wounds. It was about time for it to fly away. Before it went, it flew to the woodcutter and sang to him for the last time. After the silver bird finished, the woodcutter sighed, "Your silver feathers are very beautiful, but I think you are not as beautiful as the gold bird. Your song is very beautiful too, but still I don't think it's as beautiful as the gold bird sings."
The silver bird then flew to the direction of the setting sun. Under the golden setting sunlight, the bird became a beautiful glittering gold bird. The woodcutter then realized that the gold bird he's been dreaming for so long was really just the one beside him all the time!

Happiness does not rest on how much one possesses, but on how less one bothers. Sometimes we don't appreciate what we are having is already a fortune. We just realize (and regret to realize) only when we lose it. So appreciate what we have and don't make unnecessary comparisons.

I'm Senior

A senior citizen was driving down the M2 motorway. His car phone rang.

Answering, he heard his wife's voice saying hastily, "George, the news just said that there's a car going the wrong way on M2. Be careful!"
"My goodness!" cried George, "It's not just one car. There're hundreds of them!"

We Can Hear Just Fine

Three retirees were playing golf one fine day.

One man remarked to the other, "Windy, isn't it?"
The second man replied, "No, it's Thursday."
And the third man chimed in, "So am I. Let's have a beer."

The Pythagoras Theorem

My point of departure is the Pythagoras Theorem:
Everybody with secondary education knows the Pythagoras Theorem. But I bet over 90% of people don't know why it is true (i.e. they don't know how to prove it)! While proving the Pythagoras Theorem appears to be difficult, I present below a classical proof by Euclid (circa 330 - 260 BC) which requires no more than elementary Geometry known to every secondary student!

Please feel free to skip the rest if Geometric Proof doesn't appeal to you.
First, draw squares ABFG, BCDE and CAHK. Obviously, our task is to prove the area of square ABFG is the sum of the area of square BCDE plus the area of square CAHK. Usually people are stuck at this point. But what Euclid did was to draw a few lines as follows:

. draw CM parallel to AG, cutting AB at N and FG at M respectively
. join BH and CG

With Euclid's hints, the picture now becomes much clearer as in above figure, we can then proceed as follows:

In ΔABH and ΔAGC,
....................... AB = AG ......................... (sides of square ABFG)
.................. ∠BAH = ∠BAC + ∠CAH = ∠BAC + 90°
.................. ∠GAC = ∠GAB + ∠BAC = 90° + ∠BAC
............... ∴ ∠BAH = ∠GAC
...................... AH = AC .......................... (sides of square CAHK)
............... ∴ ΔABH Ξ ΔAGC ...................... (SAS)

....... Area of ΔABH = ½*AH*HK
........................... = ½* (AH*HK)
........................... = ½*(area of square CAHK)
....... Area of ΔAGC = ½*AG*GM
........................... = ½*(AG*GM)
........................... = ½*(area of rectangle AGMN)

Since ΔABH Ξ ΔAGC as proved above,
................ then area of ΔABH = area of ΔAGC
So, ½*(area of square CAHK) = ½*(area of rectangle AGMN)
......... ∴ area of square CAHK = area of rectangle AGMN ......... (1)

Similarly,
............. area of square BCDE = area of rectangle BFMN ......... (2)

Combining (1) and (2) together,
area of square CAHK + area of square BCDE = area of rect AGMN
................................................................... + area of rect BFMN
............................................................... = area of square ABFG

i.e. .............................................. a² + b² = c² .............. Q. E. D.