Triangle related problem

Question

In the figure below if $tan \alpha=\frac{m}{n}$, where $m$ and $n$ are relatively prime, find the value of $m$ and $n$.

My approach is as follow

$\frac{{SinA}}{{14}} = \frac{{SinB}}{{15}} = \frac{{SinC}}{{13}}$

$\Delta APB$

$\angle PBA = B - \alpha $

$\angle APB = 180 - \left( {\alpha + B - \alpha } \right) = 180 - B$

$\frac{{Sin\left( {180 - B} \right)}}{{13}} = \frac{{Sin\alpha }}{{BP}} \Rightarrow SinB = \frac{{13Sin\alpha }}{{BP}}$

$\Delta APC$

$\angle PAC = A - \alpha $

$\angle APC = 180 - \left( {\alpha + A - \alpha } \right) = 180 - A$

$\frac{{Sin\left( {180 - A} \right)}}{{15}} = \frac{{Sin\alpha }}{{AP}} \Rightarrow SinA = \frac{{15Sin\alpha }}{{AP}}$

$\Delta BPC$

$\angle PCB = C - \alpha $

$\angle BPC = 180 - \left( {\alpha + C - \alpha } \right) = 180 - C$

$\frac{{Sin\left( {180 - C} \right)}}{{13}} = \frac{{Sin\alpha }}{{PC}} \Rightarrow SinC = \frac{{13Sin\alpha }}{{PC}}$

$\frac{{SinA}}{{SinB}} = \frac{{\frac{{15Sin\alpha }}{{AP}}}}{{\frac{{13Sin\alpha }}{{BP}}}} = \frac{{14}}{{15}} \Rightarrow \frac{{{{15}^2}}}{{13 \times 14}} = \frac{{AP}}{{BP}}$

How do I approach from here?

Answer 1

P is first Brocard point of the triangle.

We have the relation $$\cot \alpha = \cot A + \cot B + \cot C$$

To quickly find these, we partition the triangle as below

so $$ \cot B = \dfrac{5}{12}, \cot C = \dfrac{3}{4} $$

and in any $\triangle$, $$ \cot A\cot B + \cot B\cot C +\cot C\cot A =1$$

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Update :

Following @JeanMarie's link, we see $$ \cot \alpha = \cot A + \cot B + \cot C = \dfrac{a^2+b^2+c^2}{4\triangle} $$

Answer 2

I am not solving the problem completely. Instead I will continue the solution from where the author has left.

Use sin property in $\Delta APB$, you will get the equation $\frac{sin(B-\alpha)}{sin(\alpha)} = \frac{AP}{BP}$. Now, use $sin(B-\alpha) = sin(B)cos(\alpha) - sin(\alpha)cos(B)$ and substitute in the previous equation to get $sin(B)cot(\alpha) - cos(B) = \frac{AP}{BP}$.

Now, you can get $cos(B) = \frac{AB^2+BC^2-AC^2}{2*AB*BC}$ using the cosine formula in $\Delta ABC$ and $sin(B) = \sqrt{1-cos^2(B)}$.

Substitute the values of $cos(B)$, $sin(B)$ and $\frac{AP}{BP}$ in the above equation and get $cot(\alpha)$. Now, $tan(\alpha) = \frac{1}{cot(\alpha)}$.