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One day Tony woke up and didn't find his wife in the house.

— Where were you?
— I wanted to work out and just walked around the block. Look at my pedometer.
— What distance does the pedometer show?
— Really Tony? again with your jealousy?

Angry, Tony snatches the pedometer from her and he read a distance traveled of $805$ meters.

— How many laps did you do?
— One lap!
— East or North?
— Are you serious, Tony?
— East or North?
— East!

Tony had been the engineer in charge of paving the block years ago, and he knew that the four streets that made up the block were the same distance (inside the block) from the church where he married his wife. In addition, he remembered that the corners formed the following sequence of angles until he got back to his house: $60°$, $135°$, $85°$, $80°$ and that if his wife went east she must have traveled $200$ meters on the first street before to reach the $60°$- corner, a distance that he also remembered perfectly.

Was the wife lying?

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  • $\begingroup$ Welcome to Puzzling, take our tour! Could you please provide proper attribution for this question? $\endgroup$
    – Varun W.
    Commented Jul 26, 2022 at 14:33
  • $\begingroup$ @VarunW., I have created this problem myself. $\endgroup$
    – Emmanuel José García
    Commented Jul 26, 2022 at 14:41
  • $\begingroup$ @VarunW., I could link to the discussion that led to this problem, but that would be a spoiler, so I'll do that later. $\endgroup$
    – Emmanuel José García
    Commented Jul 26, 2022 at 14:51
  • $\begingroup$ If it contains the answer please post it after this question has been solved. If it does not have the answer and is a hint you could put that as a hint after a while of it not being solved. $\endgroup$
    – Varun W.
    Commented Jul 26, 2022 at 14:56
  • $\begingroup$ This is the discussion that led me to create this problem. $\endgroup$
    – Emmanuel José García
    Commented Jul 27, 2022 at 16:42

1 Answer 1

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"East or north?" suggests that

the house is at a corner, so we can take the 200m as one side of the quadrilateral.

"same distance from" means

it is a circumscribed quadrilateral, in particular, the distances from each corner to the points nearest to the church are the same on either street meeting at that corner and the larger the angle, the smaller the distance. if we name the distances d60,d80,d85,d135 and the circumference C then C = 2d60+2d80+2d85+2d135 = 400m + 2d85 + 2d135 < 800m.

So that would suggest she has been

economical with the actualite.

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  • 1
    $\begingroup$ Good point @JLee. I renamed them. $\endgroup$
    – loopy walt
    Commented Jul 26, 2022 at 15:33
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    $\begingroup$ I don't see how it necessarily follows from the puzzle that that quadrilateral is circumscribed. The church could be outside of the block, as long as all four streets are tangent to a circle around the church. I am assuming of course that the four streets extend in a straight line past this block. $\endgroup$
    – Jaap Scherphuis
    Commented Jul 26, 2022 at 15:41
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    $\begingroup$ @JaapScherphuis, Thanks. I've edited the puzzle to specify that. $\endgroup$
    – Emmanuel José García
    Commented Jul 26, 2022 at 16:25
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    $\begingroup$ @JaapScherphuis I think you're right. There appears to be a configuration where C=2d80-2d135 and 200m=d80-d120. Right? $\endgroup$
    – loopy walt
    Commented Jul 26, 2022 at 16:25
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    $\begingroup$ So the quadrilateral isn't circumscribed (that would mean that it's cyclic). It's tangential. Reading that Wiki article, I see that it gives "circumscribed" as a synomym. Certainly one I would deprecate, given that it actually indicates the opposite of what it is ostensibly used to mean. $\endgroup$
    – Rosie F
    Commented Aug 2, 2022 at 15:31

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