MHB Triangle $PQR$: Find $\tan P,\,\tan Q,\,\tan R$ Values

[anemone]

In triangle $PQR$, $\tan P,\,\tan Q,\,\tan R$ are integers, find their values.

 

[kaliprasad]

In triangle $PQR$, $\tan P,\,\tan Q,\,\tan R$ are integers, find their values.

Do not ask me how to derive but the values are

Spoiler

1, 2,3

because i know

Spoiler

$\arctan(1) + \arctan (2) + \arctan (3) = \pi$

 

[MarkFL]

My solution:

Spoiler

The triple tangent identity says that for angles $x,\,y,\,z$ such that $x+y+z=\pi$, then we must have:

$$\tan(x)+\tan(y)+\tan(z)=\tan(x)\tan(y)\tan(z)$$

And so, as a consequence, we must have:

$$\left(\tan(P),\tan(Q),\tan(R)\right)$$

are one of the six permutations of:

$(1,2,3)$.

 

[RLBrown]

Do not ask me how to derive but ...

Proof#1
View attachment 2524Proof#2
View attachment 2521

 

[anemone]

Thanks all for participating!:)

Solution proposed by other:

Spoiler

Note that we have $\tan P+\tan Q+\tan R=\tan P \tan Q \tan R$.

Let $\tan P=m$, $\tan Q=n$ and $\tan R=k$, where $m,\,n,\,k$ are integers such that $m+n+k=mnk$.

We can tell $PQR$ cannot be a right triangle.

Now, suppose $\angle P$ is obtuse. Then $m$ is negative while $n$ and $k$ are positive. If $n=k=1$, then $mnk=m<m+2=m+n+k$. Any increase in the values of $n$ or $k$ will increase that of $m+n+k$ while decrease that of $mnk$. It follows that $PQR$ is an acute triangle, so that $m,\,n,\,k$ are all positive.

We may assume that $k \ge n \ge m$. Then $mnk=m+n+k \le 3k$, so that $mn \le 3$. We cannot have $m=n=1$, hence $m=1$, $n=2$, $k=3$.