perplexus.info :: Geometry : Concurrent Lines

Concurrent Lines (Posted on 2014-05-30)

Let points L, M, and N lie on the sidelines BC, CA, and AB
of ΔABC respectively. If the lines AL, BM, and CN are
concurrent at point P. Prove

      AP     AN     AM
     ---- = ---- + ----
      PL     NB     MC

, where the six line segments are directed and
the three denominators are not zero.

Submitted by Bractals

Rating: 4.0000 (1 votes)

Solution: (Hide)


Apply Menelaus' theorem to line BM and ΔACL:
AM CB LP ---- · ---- · ---- = -1 MC BL PA or AM BL PA LB PA ---- = - ---- · ---- = ---- · ---- MC CB LP CB LP AP LB = ---- · ---- (1) PL CB
Apply Menelaus' theorem to line CN and ΔABL:
AN BC LP ---- · ---- · ---- = -1 NB CL PA or AN CL PA CL PA ---- = - ---- · ---- = ---- · ---- NB BC LP CB LP AP CL = ---- · ---- (2) PL CB
Adding (1) and (2) gives
AN AM AP CL LB ---- + ---- = ---- · ( ---- + ---- ) NB MC PL CB CB AP CL + LB = ---- · ( --------- ) PL CB AP CB = ---- · ---- PL CB AP = ---- PL
QED

Note: See Harry's post for a different method.

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Solution	Harry	2014-06-02 14:46:07

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