2
$\begingroup$

Given a triangle $ABC$ with $BC = 20\sqrt{3}$ and $\angle BAC = 60^{\circ}$. The segment $AD$ is one of three altitudes of the triangle, intersects $BC$. If $H$ is an altitude point of the triangle and the area of triangle $BHC$ can be written as $a - b\sqrt{c}$ in which $a, b, c$ are natural numbers and $c$ can't be divided by square numbers except $1$, then what is $a + b - c?$

Trouble: 1. The meaning of altitude point and its position in the triangle, later to determine its area. 2. I think we should apply trigonometry, but somehow I have stucking trouble over there.

Please give me some enlightments.

$\endgroup$
2
  • $\begingroup$ Are we looking for multiple values of $a+b-c$? Since $BC$ is a chord of a circle, there are infinitely many $\triangle BHC$. $\endgroup$
    – John Glenn
    Jun 25 '18 at 16:20
  • $\begingroup$ I think the values are unique. $\endgroup$ Jun 25 '18 at 16:36
1
$\begingroup$

The area of $\triangle BHC$ is: $$A_\triangle=\frac12HC\cdot20\sqrt3\sin C$$ Since $HC=20\sqrt3\cos C$, then $A_\triangle$ becomes: $$A_\triangle=\frac12(20\sqrt3)^2\sin C\cos C=600\sin C\cos C=300\cdot2\sin C\cos C\\\implies A_\triangle=300\sin(2C)$$ Therefore, for $0<C<\frac\pi2$: $$A_\triangle=300\sin 2C=a+b\sqrt c$$ Then: $$\begin{array}{c|cc} A_\triangle&a&b&c&C\\ \hline 84-\sqrt{5}&84&1&5&\frac{1}{2} \sin ^{-1}\left(\frac{1}{300} \left(84-\sqrt{5}\right)\right)\\ 300-13 \sqrt{67}&300&13&67&\frac{1}{2} \sin ^{-1}\left(\frac{1}{300} \left(300-13 \sqrt{67}\right)\right)\\ 87-18 \sqrt{10}&87&18&10&\frac{1}{2} \sin ^{-1}\left(\frac{1}{300} \left(87-18 \sqrt{10}\right)\right)\\ 93-\sqrt{114}&93&1&114&\frac{1}{2} \sin ^{-1}\left(\frac{1}{300} \left(93-\sqrt{114}\right)\right)\\ 156-11 \sqrt{157}&156&11&157&\frac{1}{2} \sin ^{-1}\left(\frac{1}{300} \left(156-11 \sqrt{157}\right)\right)\\ 221-16 \sqrt{166}&221&16&166&\frac{1}{2} \sin ^{-1}\left(\frac{1}{300} \left(221-16 \sqrt{166}\right)\right)\\ 1300-117 \sqrt{103}&1300&117&103&\frac{1}{2} \sin ^{-1}\left(\frac{1}{300} \left(1300-117 \sqrt{103}\right)\right)\\ \vdots&\vdots&\vdots&\vdots&\vdots \end{array}$$ There are many values in between, I was only looking for values of $a,b,c$ for a unique $A_\triangle$. Hence $a+b-c$ might not be unique.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.