Ni trovis sekuran rimedon por helpi la hundon.

Because both sekuran and rimedon are in the accusative case, marked by -n.

In Esperanto, the direct object of the verb gets -n. Here, the thing that was found is rimedo (a means, method, remedy), so it becomes rimedon.

Adjectives must agree with the nouns they describe in both:

• number: singular/plural
• case: nominative/accusative

So:

• sekura rimedo = a safe means/method
• sekuran rimedon = a safe means/method as the direct object

Since trovis means found, and we found what? → a safe method, both words take the accusative ending:

• sekuran rimedon

For the same reason: la hundon is also a direct object, but this time of the infinitive helpi.

Inside por helpi la hundon:

• helpi = to help
• la hundon = the dog

You can think of it as:

• helpi кого/what? → the dog

So hundo takes -n and becomes hundon.

Por usually means for or in order to.

In this sentence, por helpi la hundon expresses purpose:

• a safe method for helping the dog
• or a safe method in order to help the dog

So por + infinitive is a very common Esperanto structure for showing purpose.

Examples:

• Mi venis por vidi vin. = I came to see you.
• Ili aĉetis ilon por ripari la pordon. = They bought a tool to repair the door.

Because after por, Esperanto normally uses the infinitive when talking about purpose.

So:

• por helpi = to help / in order to help

You would not use helpas here, because helpas is a finite verb meaning helps/is helping, and this phrase is not a full separate clause.

Compare:

• Ni trovis rimedon por helpi la hundon. = We found a method to help the dog.
• Ni helpas la hundon. = We are helping the dog.

As for helpiĝi, that would mean something very different and is not the normal form here.

Yes, rimedo can mean means, method, way, or remedy, depending on context.

In this sentence, it most naturally means:

• a means
• a method
• a solution
• possibly a remedy, if the dog is in trouble or injured

So sekuran rimedon suggests a safe way/solution rather than just a random object.

Because in Esperanto, the infinitive already ends in -i, and that ending itself carries the idea of English to.

So:

• helpi = to help
• trovi = to find
• esti = to be

You do not add a separate word for to before an infinitive in Esperanto.

Esperanto has a definite article, la, but it does not have an indefinite article like English a/an.

So:

• la hundo = the dog
• hundo = a dog / dog, depending on context

That means:

• sekuran rimedon can naturally mean a safe method
• la hundon means the dog

The speaker is referring to a specific dog, but not necessarily to a previously identified specific method.

Yes. Esperanto word order is fairly flexible because the -n ending helps show what is the object.

The standard order here is:

• Ni trovis sekuran rimedon por helpi la hundon.

But other orders are possible, for example:

• Sekuran rimedon ni trovis por helpi la hundon.
• Por helpi la hundon, ni trovis sekuran rimedon.

These may shift emphasis, but the meaning stays basically the same.

Still, the original version is the most neutral and natural for learners.

It describes rimedon, not hundon.

You can tell because:

• it is placed right before rimedon
• it agrees with rimedon in form

So the sentence means that the method is safe, not necessarily that the dog is safe.

If you wanted to describe the dog as safe, you would structure the sentence differently.

Trovis is in the past tense.

Esperanto verb endings are very regular:

• -as = present
• -is = past
• -os = future
• -us = conditional
• -u = imperative/jussive
• -i = infinitive

So:

• trovas = find / am finding
• trovis = found
• trovos = will find

That makes Ni trovis... clearly mean We found...

Normally, Esperanto includes the subject pronoun unless the subject is obvious from context or omitted for a special reason.

So:

• Ni trovis... = We found...

The verb ending tells you the tense, but not the person as specifically as in some other languages. Because of that, pronouns are usually stated.

For example:

• trovis by itself only means found
• it does not by itself tell you I found, we found, they found, etc.

So Ni is needed here to show we.