Unizor - Geometry3D - Pyramids - Problem 6

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Notes to a video lecture on http://www.unizor.com

Pyramids - Problems 6

To effectively learn from problem solving, try to solve these problems just by yourself, then listen to a lecture and then try to solve them again by yourself.

Problem A

Given a triangular pyramid SABC with apex S and side edges having sizes:
SA = a,
SB = b,
SC = c.
Inside its base ΔABC we choose any point P.
From this point P we draw three lines parallel to three side edges of a pyramid until lines intersect the opposite faces:
PA' ∥ SA where A'∈ΔSBC,
PB' ∥ SB where B'∈ΔSAC,
PC' ∥ SC where C'∈ΔSAB.
Let the length of these new segments are
PA' = a',
PB' = b',
PC' = c'.
Prove that
a'/a+b'/b+c'/c = 1