[alogo] Closed billiard paths inside a triangle

Given a triangle ABC and a line L not passing through a vertex, one can define a chain of reflexions and produce through them a billiard ball path inside the triangle, as follows:
[1] Take a side a0 intersected by L and define reflexion F0 on that side.
[2] Denote by t0 = ABC and by t1 = F0(ABC) the image of t0 under the reflexion F0.
[3] Take a side a1 of t1 intersected by L and define reflexion F1 on that side.
[4] Repeat steps [2], [3] as many times needed creating a sequence of triangles {t0, t1, t2, ...}, a sequence of sides {a0, a1, a2, ...} on these triangles and a sequence of reflexions {F0, F1, F2, ... } on these sides respectively.

[0_0] [0_1] [0_2] [0_3] [0_4]
[1_0] [1_1] [1_2] [1_3] [1_4]
[2_0] [2_1] [2_2] [2_3] [2_4]

The image above shows such a case of repeated reflexions in which line L starts at a point A of t0 and ends at a point B of t10 having the same relative position in t10 as A has in t0. The resulting billiard ball path is formed by the segments s0, F0(s1), F0*F1(s2), ... etc., where si is the segment intercepted by triangle ti on line L and (*) denotes the composition of maps.
In the above particular case the resulting billiard ball path is closed (segment d controls the distance of A from the nearest corner of t0). The path though is not periodic, meaning that the ray comming back at A is not equal inclined to the side containing A as the ray departing from A.

See Also

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