I have two tricky geometry problems begging for proofs. I have tried and failed to find classic geometry proofs for these.

1) A quadrangle ABCD satisfies the following relation on the angles:

2A + C = 360 degrees

and the following on the edge lengths:

BC = CD = DA

Prove that angle D is 60 degrees.

It is easy to see that given the three equal side lengths, and D=60, then it follows that 2A+C=360. If you change angle D then the angle 2A+C also changes and so can no longer be 360. However I don't know how to prove the opposite direction in a nice way without using trig.

I'm doing a bit of research into tiling patterns, and the above quadrangle makes quite a nice tiling, and it surprised me when I noticed that the requirements imposed on the tile by the tiling forced one angle to be constant. Here is a picture of the tiling pattern using this tile, with two different choices of side length AB.

Here is a second, and I think more difficult one I ran into.

2) A quadrangle ABCD satisfies the following relation on the angles:

A + C = 180 degrees

and the following on the edge lengths:

AB = DA

BC = CD + DA

Prove that D is 120 degrees.

I have no clue how to prove this classically, and even using trig to prove it seems to get unwieldy quickly. Is there a relatively simple proof of this?

## Two difficult geometry problems

**Moderators:** gmalivuk, Moderators General, Prelates

- eta oin shrdlu
**Posts:**442**Joined:**Sat Jan 19, 2008 4:25 am UTC

### Re: Two difficult geometry problems

For the first one:

My proof for the second one is kind of ugly right now; I want to see if it will clean up at all.

**Spoiler:**

My proof for the second one is kind of ugly right now; I want to see if it will clean up at all.

### Re: Two difficult geometry problems

For number 2 (with some trig):

**Spoiler:**

- eta oin shrdlu
**Posts:**442**Joined:**Sat Jan 19, 2008 4:25 am UTC

### Re: Two difficult geometry problems

OK, here's one for #2 with no trig:

**Spoiler:**

### Re: Two difficult geometry problems

eta oin shrdlu wrote:OK, here's one for #2 with no trig:Spoiler:

I don't follow this step:

**Spoiler:**

- eta oin shrdlu
**Posts:**442**Joined:**Sat Jan 19, 2008 4:25 am UTC

### Re: Two difficult geometry problems

jaap wrote:I don't follow this step:Spoiler:

**Spoiler:**

### Re: Two difficult geometry problems

eta oin shrdlu wrote:jaap wrote:I don't follow this step:Spoiler:Spoiler:

I may be being dense, but I'm still not seeing this.

**Spoiler:**

- eta oin shrdlu
**Posts:**442**Joined:**Sat Jan 19, 2008 4:25 am UTC

### Re: Two difficult geometry problems

jaap wrote:I may be being dense, but I'm still not seeing this.Spoiler:

**Spoiler:**

### Re: Two difficult geometry problems

eta oin shrdlu wrote:jaap wrote:I may be being dense, but I'm still not seeing this.Spoiler:Spoiler:

**Spoiler:**

It makes sense now. Thanks!

- eta oin shrdlu
**Posts:**442**Joined:**Sat Jan 19, 2008 4:25 am UTC

### Re: Two difficult geometry problems

Oops, I think I edited that sentence out... Glad you were able to decipher it anyway.jaap wrote:It makes sense now. Thanks!Spoiler:

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