Friday, September 11, 2015

Using Symmetry: More complex proof walk through

This is a continuation of the last 2 posts exploring using symmetry in proofs. This example is the most complicated yet. Symmetry via reflection produced the base of the proof but it then required recognizing an in-center to complete the process.

I started by looking at problem 45 on gogeometry.

My first instinct here was to increase the overall symmetry in the drawing by reflecting everything.
Once that's done through a variety of different angle chases you repeatedly arrive at 2b + 2a + x = 90.
The easiest place to do this is in corner BAC once the perpendicular bisector is drawn.

I could see the parallel lines in my reflected drawing which gave me a few more x angles but not another triangle equation I could use to solve the problem.

I next tried a few unsuccessful techniques. I back-cracked from the answer 30 degrees and filled in the other angles to see if any other common triangles were there that I had missed on visual  inspection.

I then removed all the x's and replaced them with  90 - 2b - 2a in the total picture to see if this helped.
I looked around to see if there was any way to use the 30-60-90 properties of side length to deduce the angle measure but that looked unlikely given the constraints.

Then I took a break for a day after hitting a wall.  And suddenly on restarting during lunch I noticed that I had the  in-center  of the triangle when redrawing the picture of one side. That meant the third side was also an angle bisector and everything fell into place. [The need for a break before finding a solution on a stumper is pretty normal for me] 

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