# Find the fallacy of the proof

• Feb 15th 2010, 06:00 AM
wvlilgurl
Find the fallacy of the proof
Discover the fallacy in the following “proof”: 45 = 60
Proof : Construct an equilateral triangle ABC. On side AB construct an isosceles right triangle ADB with AB as the hypotenuse. Construct EB on BC so that EB ~= BD. Let F be the midpoint of AD and connect E to F with a ray that extended will meet AB in a point G. Construct GD. Now construct the perpendicular bisectors of GD and GE. Because GD and GE are not parallel, the perpendicular bisectors must meet at a point K. Connect K with points G, D, E, and B.
Now, GK ~= KD and GK ~= KE (K lies on the perpendicular bisector of each of the segments GD adn GE), KD ~= KE. By construction DB ~= EB. Therefore angleKBD ~=angleKBE by SSS congruence and KBD ~= KBE. By subtraction DBG ~= EBG. But measure of angle DBG = 45 and measure of angle CBG = 60. Therefore, 45 = 60.
• Feb 15th 2010, 07:01 AM
Soroban
Hello, wvlilgurl!

Do you have a diagram or sketch?
The description has a puzzling statement.

Quote:

Discover the fallacy in the following “proof”: 45 = 60

Proof : Construct an equilateral triangle ABC.
On side AB, construct an isosceles right triangle ADB with AB as the hypotenuse.
Construct EB on BC so that EB = BD.
Let F be the midpoint of AD and connect E to F with a ray that extended will meet AB in a point G.
Why is anything extended? AB and EF intersect "internally."

Construct GD.
Now construct the perpendicular bisectors of GD and GE.
Because GD and GE are not parallel, the perpendicular bisectors must meet at a point K.
Connect K with points G, D, E, and B.

Now, GK = KD and GK = KE (K lies on the perpendicular bisector of each
of the segments GD and GE), KD = KE.

By construction DB = EB.
Therefore angle KBD = angle KBE by SSS congruence and KBD = KBE.

By subtraction DBG = EBG. **
But measure of angle DBG = 45 and measure of angle CBG = 60.
Therefore: 45 = 60.

** This is the fallacious step.

You are assuming that K, B, and G are collinear.

• Feb 15th 2010, 07:26 AM
wvlilgurl
we were told the sketch is wrong but the problem is number 4 on the following website
http://education.uncc.edu/droyster/c...homework02.pdf
• Feb 15th 2010, 07:27 AM
wvlilgurl
correction - we were told to pay attention to the sketch, it could be wonrg.