Angle bisector problem

Question

In triangle $ABC$, $AY$ is a perpendicular to the bisector $\angle ABC$ and $AX$ is a perpendicular to the bisector of $\angle ACB$. If $AB= 9cm$ , $AC=7cm$ and $BC= 4cm$, then the length of $XY$ is?

Any help would be much appreciated.

Thank you.

Answer 1

AX and AY, when both extended, cut the line BC at X’ and Y’ respectively.

Then, it is not that difficult to see that $\triangle BAX \cong \triangle BX’X$. Similarly, we have $\triangle CAY \cong \triangle CY’Y$. The consequences are:-

1) BX’ = BA, a known quantity and similarly, CY’ = CA; and

2) We can apply the midpoint theorem to $\triangle AX’Y’$ to get $XY = \dfrac {X’Y’}{2}$.

XY can then be found by observing that $X’Y’ = BX’ + CY’ – BC$.

Answer 2

Let $AB=c$, $AC=b$, $BC=a$, $\measuredangle BAC=\alpha$, $\measuredangle ABC=\beta$, $\measuredangle ACB=\gamma$,

$R$ be a radius of circumcircle of $\Delta ABC$, $S$ be an area of the triangle

and let be our bisectors intersects in the point $O$.

Thus, $$\measuredangle XAY=90^{\circ}-\frac{\beta}{2}+90^{\circ}-\frac{\gamma}{2}-\alpha=180^{\circ}-90^{\circ}-\frac{\alpha}{2}=90^{\circ}-\frac{\alpha}{2}.$$ By law of sines for $\Delta AOC$ we obtain: $$\frac{AO}{\sin\frac{\gamma}{2}}=\frac{b}{\sin\frac{\alpha+\gamma}{2}}$$ or $$AO=\frac{b\sin\frac{\gamma}{2}}{\cos\frac{\beta}{2}}.$$ But $AO$ is a diameter of the circumcircle of $AXOY$, which says $$XY=AO\sin\measuredangle XAY=\frac{b\sin\frac{\gamma}{2}\cos\frac{\alpha}{2}}{\cos\frac{\beta}{2}}=\frac{2R\sin\beta\sin\frac{\gamma}{2}\cos\frac{\alpha}{2}}{\cos\frac{\beta}{2}}=4R\sin\frac{\beta}{2}\sin\frac{\gamma}{2}\cos\frac{\alpha}{2}.$$ Now, $p=\frac{a+b+c}{2}=\frac{4+7+9}{2}=10$, which says $$S=\sqrt{10(10-4)(10-7)(10-9)}=6\sqrt5,$$ $$R=\frac{abc}{4S}=\frac{4\cdot7\cdot9}{4\cdot6\sqrt5}=\frac{21}{2\sqrt5}.$$

Now, by law of cosines for $\Delta ABC$ we obtain: $$\cos\alpha=\frac{7^2+9^2-4^2}{2\cdot7\cdot9}=\frac{19}{21},$$ $$\cos\beta=\frac{4^2+9^2-7^2}{2\cdot4\cdot9}=\frac{2}{3}$$ and $$\cos\gamma=\frac{4^2+7^2-9^2}{2\cdot4\cdot7}=-\frac{2}{7}.$$ Thus, $$\sin\frac{\beta}{2}=\sqrt{\frac{1-\frac{2}{3}}{2}}=\frac{1}{\sqrt6},$$ $$\sin\frac{\gamma}{2}=\sqrt{\frac{1+\frac{2}{7}}{2}}=\frac{3}{\sqrt{14}}$$ and $$\cos\frac{\alpha}{2}=\sqrt{\frac{1+\frac{19}{21}}{2}}=2\sqrt{\frac{5}{21}}.$$ Id est, $$XY=4\cdot\frac{21}{2\sqrt5}\cdot\frac{1}{\sqrt6}\cdot\frac{3}{\sqrt{14}}\cdot2\sqrt{\frac{5}{21}}=6.$$ Done!