Puzzle - Geometry

[imhotep]

Three identical squares. You have to use only elementary Geometry (no trigonometry) - Prove that angle C equals the sum of angles A and B.

 

[MrFrog]

Bump.
will have a look later

 

[kinkon]

 

[Stimulus mind]

 

[jamiezue]

A + α = B +γ ( sum of opposite angles = opposite exterior angle)
B +γ =C (corresponding angles )
A + α = C
A+B +γ +α = 2C
C= x+y+D
x+D= α (alternate interior angles )
C= y+α
y=A
to proove A+B=C,>>>> γ +α =C
but γ +α not equal to C per 2 nd square . Because its already more than 45 degrees . So I dont know.....This is so far..below is the sketch. I added the stuff in REd color

 

[imhotep]

A + α = B +γ ( sum of opposite angles = opposite exterior angle)
B +γ =C (corresponding angles )
A + α = C
A+B +γ +α = 2C

Tip: You will obviously need an additonal construction done to the existing figure as a Geometric proof is requested.

Otherwise quite simply A + B = tan^-1(1/3)+tan^-1(1/2) = π/4 = C

Additonally you could assume that it's in the complex plane and using the i notation prove the same thing. But what's needed is a pure Geometric proof.

 

[Solo Rider]

 

[nlasasatha]

 

[imhotep]

View attachment 230706

Nice... Very good effort. This is one of the solutions.

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Here are the solutions. If anyone finds others please post..

Geometrical Solutions. - The first is the one @nlasasatha posted.

1)



2) Alternate Geometric




Non-Geometrical Solutions

3)



Imagine the squares are in the complex plane, Origin, the lower left hand vertex. Horizontal the real axis and Vertical i axis. Let Z1, Z2 and Z3 represent three complex numbers in an Argand diagram.
Z1 = 3+i, Z2 = 2+i and Z3 = 1 +i,

Our angles of interest are the "Arguments" of these three complex numbers. Now we know that if we multiply complex numbers then the Arguments gets added.
I leave it to you to multiply Z1, Z2 and Z3. You will get the answer 10i. So the argument of the multiplication of the three numbers is π/2.
Thus A + B + C = π/2

4) Trigonometry

The other solution is the one I already mentioned above using Trignometry and the property of the addition of two inverse tangents.
A + B = tan^-1(1/3)+tan^-1(1/2) = π/4 = C
------ Post added on Jun 2, 2024 at 5:57 AM