Exercise 10.3.14

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

As explained in [21](C.R.Videla, {\it On points constructible from conics}), Pappus used intersections of conics to trisect angles as follows. Consider the unit circle centered at the origin, and let $0<	heta < \pi/2$. Then $P = (\cos 	heta, \sin 	heta)$ is the corresponding point on the unit circle. We assume that $P$ is known. Also let $O =(0,0)$ and set $A = (1,0)$. Thus $	heta = \angle POA$.
\be
\item[(a)] Consider the curve $C$ consisting of all points $Q = (x,y)$ such that the distance from $P$ to $Q$ is twice the distance from $Q$ to the $x$-axis. The curve $C$ intersects the unit circle at a point $R$ lying in the interior of $\angle POA$. Prove that $\angle ROA = 	heta/3$.
\item[(b)] Show that the curve $C$ is a hyperbola. It follows that we have trisected an angle using the intersection of a hyperbola and a circle, i.e., an intersection of conics.
\ee

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}% intervalles d'entiers 
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\be} {\begin{enumerate}}

ewcommand{\ee} {\end{enumerate}}

ewcommand{\deb}{\begin{eqnarray*}}

ewcommand{\fin}{\end{eqnarray*}}

ewcommand{\ssi} {si et seulement si }

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{\U}{\mathbb{U}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\im}{\,\mathrm{Im}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{\Gal}{\mathrm{Gal}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof}
\be
\item[(a)]
Let $H$ be the orthogonal projection of $R$ on the $x$-axis, and $S$ such that $H$ is the midpoint  of $RS$, so $S$ is the reflection of $R$ with regard to the $x$-axis. Since $R$ is on the curve $C$,
$$PR = 2 RH = RS,$$
therefore the measures of $\angle ROP$ and $\angle SOR$ are equal, and is twice the measure of $\angle AOR$, so $\overrightarrow{OR}$ trisects the angle $\angle POA$ and a measure of $\angle ROA$ is $	heta/3$.

\item[(b)] Let $Q=(x,y), H = (x,0), P = (\cos 	heta, \sin 	heta)$
\begin{align*}
Q \in C &\iff QP = 2 QH \iff QP^2 = 4 QH^2\
&\iff(x- \cos 	heta)^2 + (y-\sin 	heta)^2 = 4 y^2\
& \iff x^2 - 3 y^2  - 2 x \cos 	heta  - 2 y \sin 	heta +1 = 0.
\end{align*}
The discriminant of the quadratic form $a x^2 + bxy + cy^2 = x^2-3y^2$ is $\Delta = 12>0$, so $C$ is not empty and $C$ is a hyperbola.
\ee
\end{proof}

Exercise 10.3.14

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Exercise 10.3.14

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