A new(?) proof of the Law of Cosines using segments determined by the points of contact with a triangle's incircle
In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using standard notation, the law of cosines states
$$c^2=a^2+b^2-2ab\cos{\gamma}$$
where $\gamma$ denotes the angle contained between sides of lengths $a$ and $b$ and opposite the side of length $c$. The other two relations are analogous:
$$a^2=b^2+c^2-2ac\cos{\alpha}$$$$b^2=a^2+c^2-2ac\cos{\beta}$$
Proof. Let $D$, $E$ and $F$ be the contact points of the incircle with $AC$, $AB$ and $BC$, respectively. Also, let $AE=AD=x$; $BE=BF=y$; $CD=CF=z$. We start from two well-known relationships of a triangle: $$\sin^2{\frac{\gamma}{2}}=\frac{(s-a)(s-b)}{ab} \qquad\text{and}\qquad \cos^2{\frac{\gamma}{2}}=\frac{s(s-c)}{ab} \tag{1}$$ (See Cut-the-Knot's "Relations between various elements of a triangle" page for proofs.) Here, $s$ denotes the semiperimeter of $\triangle{ABC}$. Since $(s-a)=x$, $(s-b)=y$ and $(s-c)=z$, then the following identity holds:
$$xy\left[\frac{s(s-c)}{ab}\right]=sz\left[\frac{(s-a)(s-b)}{ab}\right]\tag{2}$$
Now, rewriting, $$\begin{align} xy\cos^2{\frac{\gamma}{2}}&=sz\sin^2{\frac{\gamma}{2}} \tag{3} \\[4pt] xy(1+\cos{\gamma}) &=sz(1-\cos{\gamma}) \tag{4} \\[4pt] xy(1+\cos{\gamma}) &=(xz+yz+z^2)(1-\cos{\gamma}) \tag{5} \end{align}$$Expanding, multiplying by 2, rearrenging and factoring, $$2xy=2yz+2xz+2z^2-2(y+z)(x+z)\cos{\gamma} \tag{6}$$Adding $x^2+y^2$ to both sides, $$\begin{align} x^2+2xy+y^2 &=(y^2+2yz+z^2)+(x^2+2xz+z^2)-2(y+z)(x+z)\cos{\gamma} \tag{7} \\[4pt] (x+y)^2 &=(y+z)^2+(x+z)^2-2(y+z)(x+z)\cos{\gamma} \tag{8} \\[4pt] c^2 &=a^2+b^2-2ab\cos{\gamma} \tag{9} \end{align}$$As desired. $\square$
A similar reasoning must show that $a^2=b^2+c^2-2bc\cos{\alpha}$ and $b^2=a^2+c^2-2ac\cos{\beta}$.
Is this new?
$\endgroup$ 2 Reset to defaultKnow someone who can answer? Share a link to this question via email, Twitter, or Facebook.
More in general
"Zoraya ter Beek, age 29, just died by assisted suicide in the Netherlands. She was physically healthy, but psychologically depressed. It's an abomination that an entire society would actively facilitate, even encourage, someone ending their own life because they had no hope. Th…"
