Two identically oriented equilateral triangles,
![ABC](/media/m/a/c/7/ac75dca5ddb22ad70f492e2e0a153f95.png)
with center
![S](/media/m/c/6/3/c63593c3ec0773fa38c2659e08119a75.png)
and
![A'B'C](/media/m/f/d/7/fd762a86f4223d7dedfc3eb57b9d2959.png)
, are given in the plane. We also have
![A' \neq S](/media/m/6/3/e/63ea1f7d0439d7e98ba809d3162241d5.png)
and
![B' \neq S](/media/m/5/f/3/5f34cca78391990d40b8940787ede48f.png)
. If
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
is the midpoint of
![A'B](/media/m/c/1/1/c11fcf5ec02b69448b3dc3eda929a93b.png)
and
![N](/media/m/f/1/9/f19700f291b1f2255b011c11d686a4cd.png)
the midpoint of
![AB'](/media/m/3/c/e/3cee1291aba73d23886dc1ec0bb29387.png)
, prove that the triangles
![SB'M](/media/m/e/e/7/ee796b0c3db1fdde3c670ccdc1ba4ddc.png)
and
![SA'N](/media/m/6/9/5/695ab9b6588ff97501946114bfe1afdf.png)
are similar.
%V0
Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A' \neq S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.