# The Problem of Triangle in Geometry

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.

• Are we looking for multiple values of $a+b-c$? Since $BC$ is a chord of a circle, there are infinitely many $\triangle BHC$. Jun 25 '18 at 16:20
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.