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Two Parallels in a Triangle and One More: What Is It About?
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Copyright © 1996-2010 Alexander Bogomolny
Two Parallels in a Triangle and One More
The applet illustrates the following problem
Let X be a point on the side BC of a triangle ABC. The parallel through X to AB meets CA at V and the parallel through X to AC meets AB at W. Let
The problem has been proposed Francisco Javier García Capitán, Spain for Mathematical Reflections (2, 2009).
Two pairs of intersecting lines (AC, CW) and (CW, BC) are crossed by two parallel lines AB and VX. This gives
implying
On the other hand, two parallels WX and AC cross AB and BV. This gives
By transitivity,
It follows that indeed DE||BC. Let P be the intersection of AB and DE. Since PE||BX and EX||BP, BXEP is parallelogram and
or,
which can be rewritten as
And, finally, dividing by BX, we get
as required.
Copyright © 1996-2010 Alexander Bogomolny |
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