Problem 663. Scalene Triangle, 60 Degrees, Altitude, Angle Bisector, Perpendicular Bisector, Concurrency. Level: High School, College.

The figure shows a scalene triangle ABC with angle B 60 degrees. Prove that altitude AD, angle bisector BE, and perpendicular bisector FG are concurrent at a point M.


  
 

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 ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."
 

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?