I am trying to understand the solution of 80-20-20 triangle problem as stated here http://www.cut-the-knot.org/triangle/80-80-20/Classical4.shtml#Knop
One thing I cannot get understand from the solution is how BP = CD. Pointers on this is appreciated. The rest of the solution is fine.
You know that $\angle CBD = 50$ degrees and $\angle ACB = 80$ degrees, so that gives you $\angle BDC=50$ degrees.From this, you know that $\triangle BCD$ is isosceles, so $BC=CD$ and hence $BP=CD$. Then, $P$ is constructed as the third vertex of an equilateral triangle with base $BC$, so $BP=BC$ and so $BP=BC=BD$.