MathLabs

Problem 2

A non-isosceles triangle A1A2A3A_1A_2A_3 has sides a1a_1, a2a_2, a3a_3 with the side aia_i lying opposite to the vertex AiA_i. Let MiM_i be the midpoint of the side aia_i, and let TiT_i be the point where the inscribed circle of triangle A1A2A3A_1A_2A_3 touches the side aia_i. Denote by SiS_i the reflection of the point TiT_i in the interior angle bisector of the angle AiA_i. Prove that the lines M1S1M_1S_1, M2S2M_2S_2 and M3S3M_3S_3 are concurrent.
Step 5 of 8: Pairwise parallel sides mean the two triangles are homothetic or translates
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In plain words

Parallelism of all three sides is a strong rigidity condition: it leaves only scaling-from-a-point or sliding as possibilities, nothing else.

△S1S2S3 and △M1M2M3 have pairwise parallel sides ⇒ homothetic or translates\triangle S_1S_2S_3 \text{ and } \triangle M_1M_2M_3 \text{ have pairwise parallel sides} \ \Rightarrow\ \text{homothetic or translates}
Detailed analysis

It is a standard fact that if two triangles have their three pairs of corresponding sides respectively parallel, then one is the image of the other under either a homothety (possibly with a negative ratio) or a translation. Step 4 establishes exactly this hypothesis for S1S2S3S_1S_2S_3 and M1M2M3M_1M_2M_3, with correspondence Si↔MiS_i\leftrightarrow M_i.