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=?

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

 

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...

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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

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, 20232 yr by md65536

2 minutes ago, md65536 said:

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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  ).