World's Hardest Easy Geometry Problem

I wasted a lot of time yesterday with this geometry problem.  I have about 12 pieces of paper here that look like a Mondrian retrospective, cutting new triangles and parallel lines.  Still don't have the proof yet, so I thought I would see if I could pull some of your productivity down with mine.  If you are like me, you will decide that the answer is trivial about twice in the first five minutes, both times discovering you have not actually gotten to the answer.


  1. Michael H.:

    It took me about 20 minutes. But the trick is to understand that if things were not just right, angle x would have to be a complex trigonometric formula. Work backward from this hint.

  2. delurking:

    Michael H. wrote:
    "It took me about 20 minutes. But the trick is to understand that if things were not just right, angle x would have to be a complex trigonometric formula. Work backward from this hint."

    Right. The biggest hint on the page is the fact that the "second hardest easy geometry problem" is actually distinct from the first one. If you try solving it for arbitrary angles, you get nowhere.

  3. Simon Allaway:

    Once I filled in a few blanks and realized the bottom two angles were the same it was easy. 15 minutes.

  4. Dan:

    I must have solved it differently. Given the included information, there were only four angles, including x, that were not directly calcuable. From there I created a system of four equations with four variables to solve for the remaining angles.

    20 minutes here as well.

  5. Dan:

    Other guy named Dan:

    I'm pretty sure you have an error. I too found all but four of the angles. Unfortunately, the system of equations had one redundant equation, so it wasn't sufficient. Maybe you found a different fourth equation that I didn't see.

  6. nicole:

    Did the same as the first Dan and every way I can think of to check my work it comes out right.

    Of course, that doesn't mean it isn't wrong. My own reaction is that I must have an error because this only took me about five minutes. Not sure what you mean by redundant equation though.

  7. TJIC:

    I tried the four simultaneous equations approach last night...and I found what the second Dan found: there wasn't enough there to solve the equations (or "the equations had an infinite number of solutions").

  8. Michael H.:

    I have to admit that it seemed to me later it was too easy and I was right about that - I had it wrong. So I decided to cheat and use the law of sines because I wanted to know the answer. It was really interesting and very unexpected. I didn't give me any intuition in solving the problem at all. It probably is really difficult. I see similar triangles but no congruent ones. It may require constructions "outside the box". It is a classic alright.

  9. AC:

    For the curious, there is a full solution here. Not something I would have arrived at anytime soon.

  10. Andy:

    Thanks for the time waster! :) It's been a dog's age since I used geometry. Amazingly, I recalled most of the rules, but got hung up at creating congruent angles. I even bisected Angle C, before stalling out. It all made sense, after I peeked at AC's cheat.

    Definitely a keeper for when my kids get into geometry.

  11. Jim Collins:

    Took about 15 seconds. Of course I took a drafting machine and layed out the angles, then measured with a protractor.

  12. Ivan:

    I did it (hardest problem), using only basic geometric laws and one system with 3 unknown variables at the end. But it took my almost 2 hours.

  13. Ivan:

    Ac, i think solution you linked is wrong. But i have to check it out once more.

  14. Ivan:

    Sorry, solution is right, I made a calculation error previously. I have found solution much easier than guy in the link.

  15. Bryan Pick:

    Lucky I have so much free time. I spent hours on this -- but only used a post-it note and one sheet of printer paper -- and then left it to sleep on it, quite sure I was only a step away from the answer but still unhappy.

    Turns out I was right, and I felt foolish for not seeing earlier the pair of matched angle measures that led me to the solution. I did it much the same as the linked proof, only without formally writing each step of the proof (and I didn't feel the need to draw line HF, although I drew a few other, extraneous lines). I had even intuited the final answer correctly.

  16. Bryan Pick:

    Err, I didn't originally feel the need to draw HF.

  17. Corky Boyd:

    I came up with x=60, angle EDB=70 and the intersection of BD and AE=50. Had to use simultaneous equations to figure x. Didn't look at the clues. Is this correct?

    Has been over 50 years since I took geometry.

  18. RC:

    Sorry, but x=50's not right.

  19. Mike Coffin:

    It took me several hours, but that's ok --- I did it on company time :-)

    Follow my link if you're curious.

  20. Rajit:

    x=20, took about 15 min.

  21. Rajit:

    x=20, took about 15 min.

  22. LoomingLoomer:

    I am 10, and this is not so hard, but it took me 1 hour, and my answer is 20 degrees.

  23. Jace:

    1 minute... it was not hard at all. x=20. im a sophomore in high school. I dont know why everyone thinks it is so hard

  24. mohamed:

    please someone send me the answer and how they find it it took me 5 hours and i get nothing send it plz send it to