Problem 689: Triangle, Three Excircles, Tangency points, Tangent lines, Concurrent Lines,
Mind Map, Polya.
Level: High School, College, SAT Prep.
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.
Geometry problem solving
Geometry problem solving is one of the most
challenging skills for students to learn. When a
problem requires auxiliary construction, the
difficulty of the problem increases drastically,
perhaps because deciding which construction to
make is an illstructured problem. By
“construction,” we mean adding geometric figures
(points, lines, planes) to a problem figure that
wasn’t mentioned as "given."
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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