2010 USAMO Problems/Problem 4
Problem
Let be a triangle with
. Points
and
lie on sides
and
, respectively, such that
and
. Segments
and
meet at
. Determine whether or not it is possible for
segments
to all have integer lengths.
Solution
We know that angle , as the other two angles in triangle
add to 45^{\circ}
AB, BC, BI
CI
BC^2 = BI^2 + CI^2 - 2BI*CI*cos 135^{\circ}
BC^2 = AB^2 + AC^2
cos 135^{\circ} = -\frac{\sqrt{2}}{2}
AB^2 + AC^2 - BI^2 - CI^2 = BI*CI*\sqrt{2}$$ (Error compiling LaTeX. Unknown error_msg)\sqrt{2} = \frac{AB^2 + AC^2 - BI^2 - CI^2}{BI*CI}
\sqrt{2}
\sqrt{2}
AB, BC, BI
CI$ to all be integers, which invalidates the original claim that all six lengths are integers, and we are done.