Genady Posted May 2, 2023 Share Posted May 2, 2023 Link to comment Share on other sites More sharing options...
Genady Posted May 6, 2023 Author Share Posted May 6, 2023 To ease the discussion, let's name all the important pieces: Given: a+b+c=6, d+e=3, f+g+j=5. The question is: a+i+h+j=? Link to comment Share on other sites More sharing options...
md65536 Posted May 12, 2023 Share Posted May 12, 2023 I see there's other ways to figure this out, but I noticed that there are lots of ways the DE line can be chosen... Spoiler And if you maximize the length DC, EC goes to 0 and you get a degenerate triangle with perimeter 2 DC. But you can also maximize EC and get perimeter 2 EC. But if any DE line works, those maximums would have to be the same length... does that always happen in general? Anyway the answer I get is Spoiler 8 1 Link to comment Share on other sites More sharing options...
Genady Posted May 12, 2023 Author Share Posted May 12, 2023 54 minutes ago, md65536 said: I see there's other ways to figure this out, but I noticed that there are lots of ways the DE line can be chosen... Hide contents And if you maximize the length DC, EC goes to 0 and you get a degenerate triangle with perimeter 2 DC. But you can also maximize EC and get perimeter 2 EC. But if any DE line works, those maximums would have to be the same length... does that always happen in general? Anyway the answer I get is Hide contents 8 The answer is right, and it is a very good heuristic, but it is not rigorous. It doesn't happen to be so in general - it works here because we assume that the answer is completely determined by the given data. +1 Link to comment Share on other sites More sharing options...
md65536 Posted May 12, 2023 Share Posted May 12, 2023 (edited) 37 minutes ago, Genady said: It doesn't happen to be so in general - it works here because we assume that the answer is completely determined by the given data. Spoiler I think it does generalize, and that the answer is completely determined by the data because it generalizes. Or to put it another way, a+b = g+j for any inscribed triangle, regardless of the other data. The generalization is that if 2 intersecting lines are both tangent to a circle, the intersection point is equidistant to the 2 tangent points. I used that equality about 4 more times to solve it. Edited May 12, 2023 by md65536 Link to comment Share on other sites More sharing options...
Genady Posted May 12, 2023 Author Share Posted May 12, 2023 2 minutes ago, md65536 said: Reveal hidden contents I think it does generalize, and that the answer is completely determined by the data because it generalizes. Or to put it another way, a+b = g+j for any inscribed triangle, regardless of the other data. The generalization is that if 2 intersecting lines are both tangent to a circle, the distance from the intersection point is equidistant to the 2 tangent points. I used that equality about 4 more times to solve it. Yes, you are right (I know how you did it ). Link to comment Share on other sites More sharing options...
Recommended Posts
Create an account or sign in to comment
You need to be a member in order to leave a comment
Create an account
Sign up for a new account in our community. It's easy!
Register a new accountSign in
Already have an account? Sign in here.
Sign In Now