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.

  • $\begingroup$ 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? $\endgroup$ – Blue Oct 8 '17 at 21:23
  • $\begingroup$ @Blue Yes, sorry I was unclear in my post. Your assumptions are correct. $\endgroup$ – 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.


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