Consider an equilateral triangle $ABC$ of side $l$, and a point $P$ inside $ABC$.
The following picture represents a proof without words of the fact that, given any $P$, the sum of the segments $x,y,z$ is invariant for any $P$, and $x+y+z=l$.
The segments $x,y,z$ are obtained by letting three lines, each parallel to one of the sides, passing through the point $P$.
Similarly, the following picture shows that, given any $P$, also the sum of the lengths of the segments $\alpha,\beta,\gamma$ is invariant, and such that $\alpha+\beta+\gamma=l$.
The segments $\alpha,\beta,\gamma$ are obtained by carrying the shorter sides of the triangle $APC$ on its longest one.
How can we map each $(x,y,z)$ tuple to each $(\alpha,\beta,\gamma)$ tuple (and vice versa)?
NOTE: This post is the corrected version of a previous one, with similar title, that I deleted because of errors pointed out by clever and kind users. Thanks!