A semicircle and an interior circle are tangent to each other. The chord towards the tangency point is divided in segments of length 1 and 2. What is the pink area?

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## Solution

The pink area is (2-√3)(7π+3)/12, which is approximately 0,56.

The proof starts by using the fact that for interior touching circles, radii and chords are aligned. Therefore triangles GCE and GHB are similar and have length proportion 1 : 2. Thus the semicircle has three times the radius of the circle.

Now from the Inscribed angle theorem and from sin(2α)=1/2, we find that 2α=30°. Therefore β=150°.

Focussing on triangle GCE, it is straightforward to find that r=1/(2sin(75)). The pink area is a circular sector of 210° plus triangle GCE, which amounts to r^{2}(21π/36+sin(150)/2). This can be worked out to the expression above.

## Poems

The sleeping baby

So tender in the night

The parents are caring

Suddenly a bird is twittering

A nice song

I remember my childhood

Flashback to my past

My mother the cradle rocking

And sorrows comforting

And a gently rocking cradle for a peaceful resting.

Day after day

Is coming the night.

While bells are ringing,

We’re on time and fight

Every day

Life is so good

But do run to fast.

While it’s so stunning,

Expect the forecast:

Sky will be good.