# In an equilateral triangle RST, points K, L, M are located on the sides such that RK=SL=TM. Prove that triangle KLM is equilateral.

In an equilateral triangle RST, points K, L, M are located on the sides such that RK=SL=TM.

Prove that triangle KLM is equilateral.

I have been trying to conduct this proof, but I can't seem to figure out how to show that KLM is equilateral. So far I understand that since triangle RST is equilateral, RK=SL=TM and SK=LT=MR. I have been trying to figure out how to use ASA or SAS congruence to prove the statement. Any help would be greatly appreciated.

• The description is unclear as to where exactly $K$, $L$, $M$ lie. Your discussion suggests that $K$ is on $\overline{RS}$, $L$ is on $\overline{ST}$, and $M$ is on $\overline{TR}$. Is that your intention? – Blue Oct 8 '17 at 21:23
• @Blue Yes, sorry I was unclear in my post. Your assumptions are correct. – Katiee Oct 8 '17 at 21:27

You are almost there. Observe that $\triangle KSL \cong \triangle LTM \cong \triangle MRK$ through SAS rule as
• Side: $\overline{KS} = \overline{LT} = \overline{MR}$
• Angle: $\angle KSL = \angle LTM = \angle MRK = 60 ^\circ$
• Side: $\overline{SL} = \overline{TM} = \overline{RK}$
Therefore, $\overline{KL}=\overline{LM}=\overline{MK} \implies \triangle{KLM}$ is equilateral.