If the slopes of bisectors are rational numbers, the slopes of sides are also rational numbers

Let $$m_1, m_2, m_3$$ be the slopes of the cartesian equations of the sides of a triangle of which no side is parallel to y-axis, and let $$s_1,s_2,s_3$$ be the slopes of the cartesian equations of the internal angle bisectors of the same triangle. The axes are orthogonal. Then we may say:

If $$s_1,s_2,s_3$$ are rational numbers, then $$m_1, m_2, m_3$$ are also rational numbers.

Any hints? This theorem seems a bit counterintuitive, because we cannot state the reverse or reciprocal of it.

• Isn't the slope of an angle bisector the average of the slopes of the two rays of the angle? – fleablood Sep 26 '18 at 19:15
• Well, $\tan \theta \in \mathbb Q\implies \tan 2\theta\in \mathbb Q$ so it isn't that odd. – lulu Sep 26 '18 at 19:16
• @fleablood Oh, no. look at the angle formed by the lines $y=x$ and the $x-$axis. The angle bisector of that doesn't even have rational slope...indeed $\tan \frac {\pi}8=\sqrt 2 -1$ – lulu Sep 26 '18 at 19:19
• @fleablood the slope of the angle bisector won't just be the mean of the slopes of the two rays of the angle. For example, if one ray had a slope of 0, and the other had a slope of 1,000, you would nearly have a right angle. But averaging those slopes you get 500, which is also very steep. The bisector of such an angle would have a slope just a tiny bit less than 1. – Kurt Schwanda Sep 26 '18 at 19:19
• " This theorem seems a bit counterintuitive, because we cannot state the reverse or reciprocal of it." That's your critereon for "intuitive"? But being able to state the reverse or reciprical (are those the words you want) has nothing to do with whether the statement is true or not. But in't the negation "it's possible for $s_i$ to be rational while some $m_i$ are irrational" and isn't the contrapositive "if any of the $m_i$ are irrational then one or more of the $s_i$ is irrational"? – fleablood Sep 26 '18 at 19:20

$$\tan(2t) =\dfrac{2\tan(t)}{1-\tan^2(t)}$$ so if $$\tan(t)$$ is rational so is $$\tan(2t)$$.
• Ok. If $s_1,s_2,s_3$ are rational numbers, the tangents of the angles of the triangle are also rational numbers, I can see your point. But does this finding allow us to say that $m_1,m_2, m_3$ are also rational numbers? – MrDudulex Sep 26 '18 at 23:23