Pembuktian Teorema Menelaus

Soal berikut disajikan dalam Final Contest SMU atau MC.SHS di FIM 2010 di Jurusan Matematika Fakultas MIPA Unnes dalam rangkaian September MIPA 2010.

Dipunyai segitiga ABC seperti tampak pada gambar di bawah ini!
Sebuah garis melintas membagi segitiga tersebut menjadi 2 bangun, memotong sisi segitiga dan perpanjangannya di P, Q, dan R.

Buktikan

Penyelesaian!
Draw segment a, b, and c, such that a // b // c. Line a pass through vertex A, line b pass through vertex B, and also line c. See figure below!

Proofing of Menelaus theorem

Look at triangle PAK and triangle PBM!
Obvious triangle PAK ≈ triangle PBM (Why?). So that we get
Because P doesn’t lie between A and B, then we get

Look at triangle QBM and triangle QCL!
Obvious triangle QBM ≈ triangle QCL (Why?) So that we get

Because Q lies between B and C, then we get

Look at triangle RAK and triangle RLC!
Obvious triangle RAK ≈ triangle RLC (Why?) So that we get

Because R lies between C and A, then we get

Thus, from three equations above we get

This proves Menelaus theorem