Problem 493. Right Triangle, Circumcircle, Perpendicular, Chord.
Level: High School, College, SAT Prep.

The figure shows a right triangle ABC (angle B = 90 degrees) with the circumcircle C₁ and center O. From point D on C₁, perpendiculars DE and DF are drawn to AC and AB, respectively. Lines BD and EF intersect at G. Prove that OG is perpendicular to BD.
Circumcircle of a triangle is the circle which passes through the vertices.

 

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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