Problem 689: Triangle, Three Excircles, Tangency points, Tangent lines,
Concurrent Lines
The figure shows a triangle ABC with
the excircles E_{A}, E_{B}, and E_{C} tangent to the
extension of sides at the points
A_{1}, A_{2}, B_{1}, B_{2}, C_{1},
and C_{2}. Lines BC_{1} and CB_{2} meet at A_{3},
CA_{1} and AC_{2} meet at B_{3}, AB_{1}
and BA_{2} meet at C_{3}. Prove that lines A_{3}A,
B_{3}B, and C_{3}C are concurrent.
How to Solve It, Interactive Mind Map
George Pólya's 1945 book "How to Solve It, A new
aspect of Mathematical Method", is a book
describing methods of problem solving. It
suggests the following steps when solving a
mathematical problem: (1) First, you have to
understand the problem. (2) After understanding,
then make a plan. (3) Carry out the plan. (4)
Look back on your work. How could it be better? Mind Map Help. To see a note: Hover over a yellow note button. To Fold/Unfold: click a branch. To Pan: click and drag the map canvas. 
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