Problem 506. Triangle with Three Squares, Center, Midpoint, Perpendicular, Congruence. Level: High School, College, SAT Prep.

The drawing shows a triangle ABC with the squares ABDE, BCGH and ACPQ. M is the midpoint of EG and O is the center of the square ACPQ. Prove that: (1) AMCO is another square, (2) BO and MN are congruent and perpendicular.

 

 
  

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? 
 

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