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Steam locomotive puzzle
  The 3 wheels with center O₁, O₂ and O₃ are of the same size and tangent to each other. TO₃O₁ is a right triangle. If the radius of the wheels is 6 cm long what is then the length of the segment AB?

Difficulty level: , general math knowledge.
Category: Geometry.
Keywords: tangent, circle, radius.
Related puzzles:
- A mathematic shield,
- Achtung minen.

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AB is a chord of the circle with center O₂
MO₂ is perpendicular to AB. In a circle, a radius perpendicular to a chord bisects the chord (and the arc). Thus, AM = MB = AB/2

From the similarity of triangles TO₁O₃ and MO₂O₃ :
MO₂ / TO₁ = O₂O₃ / O₁O₃ = 1/2
MO₂ = 1/2 x 6 = 3 [cm]

Applying Pythagorean theorem on the triangle MO₂B :
MB = √(6² - 3²) = 3√3
AB = 2 x MB = 6√3 [cm]

Geometric definitions
Tangent: In classical geometry, the tangent line (or simply the tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point.
Radius: In classical geometry, the radius of a circle or sphere is any line segment from its center to its perimeter.
Chord: In classical geometry, a chord is a geometric line segment whose endpoints both lie on the circumference of the circle.

The 5 Winners of the Puzzle of the Month are:
Serhat Duran, Turkey - Evis Hoxa, Albania - Ramnarayan Panda, India - Marlon Manto, USA - Shashank Rathore, India

Congratulations!

A problem, sometimes known as Moser's circle problem, asks to determine the number of regions into which a circle is subdivided if n points on its circumference are joined by chords. The answer is:
  (n⁴ - 6n³ + 23n² - 18n + 24)/24

The first values are then 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, 386, 562, ... (This is often given as an example of what happens if you attempt to guess a sequence from the first few terms since this sequence starts with 1, 2, 4, 8, 16, but the next term is 31 and not 32 as expected)

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