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1% fig-iii-36.tex — III.36: power of a point (secant and tangent).
2\begin{figure}[H]
3\centering
4\begin{tikzpicture}[scale=1.1, line cap=round]
5 \coordinate (O) at (0, 0);
6 \def\r{1.8}
7 \draw[thin] (O) circle (\r);
8 % External point P.
9 \coordinate (P) at (4.0, 0);
10 % Secant through P meeting circle at A (near) and B (far).
11 \coordinate (A) at ({\r*cos(120)}, {\r*sin(120)});
12 \coordinate (B) at ({\r*cos(45)}, {\r*sin(45)});
13 % Line through A and B extended to P (P is constructed beyond B).
14 % Pick A and B on the circle, then place P on line AB extended.
15 % For visual clarity, just draw P--A--B.
16 \draw[very thick] (P) -- ($(B)!1.6!(A)$);
17 % Tangent from P touching circle at T.
18 % T is found by: OT perpendicular to PT, and |OT|=r, |OP|=PO.
19 % Compute T: angle OPT = arcsin(r/OP).
20 \pgfmathsetmacro{\OPdist}{4.0}
21 \pgfmathsetmacro{\angA}{asin(\r/\OPdist)}
22 \coordinate (T) at ({(\OPdist*cos(\angA))*cos(180 - \angA)}, {(\OPdist*cos(\angA))*sin(180 - \angA)});
23 \draw[very thick] (P) -- (T);
24 \draw[thin, dashed] (O) -- (T);
25 % Right angle marker at T (small square).
26 % Labels.
27 \node[right] at (P) {$P$};
28 \node[above left] at (A) {$A$};
29 \node[above right] at (B) {$B$};
30 \node[above] at (T) {$T$};
31 \node[below] at (O) {$O$};
32\end{tikzpicture}
33\caption{Proposition III.36. From an external point $P$, the tangent
34$PT$ and any secant $PAB$ satisfy $PT^2 = PA \cdot PB$ (the power of
35$P$ with respect to the circle).}
36\label{fig:III.36}
37\end{figure}
38