The merging triangle

{The Merging Triangle}
The merging triangle is a famous puzzle dealing with the kinematics of a symmetric situation. The statement of the problem is as follows:

The vertices of an equilateral triangle ABC of length d start moving with constant speed u such that:

Question: After how much time do the 3 points meet?

One must first convince himself that the 3 points do meet! And after that, we should think where do they meet? The symmetry of the problem is such that the points will meet at the centroid or better barycenter of the triangle (the point where the three heights intersect). Hence we can prove that the barycenter of the triangle has no net speed and is the only fixed point in our situation!

At any moment, the points
will form an equilateral triangle with smaller side length. So:

Then imagine that we are sitting at the center of the triangle. How are we seeing the points moving toward us? We surely see them approach us but also rotate around us. The net velocity of the two is their speed u which is constantly changing direction. The rotational speed and the radial speed form a triangle similar to “half” our equilateral. Hence geometry will solve our problem!

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