# Quod Erat Demonstrandum

## 2017/01/10

### 某經典幾何題

Filed under: NSS,Pure Mathematics — johnmayhk @ 5:03 下午
Tags: , ,

https://www.geogebra.org/m/WPk7sZUJ

（若有興趣知如何構作兩圓的外公共切線，可看文末的附錄*）

//He paused for a moment and said, “Yes, that is perfectly self-evident." Astonished, his friend asked him to explain … Prof. Sweet, in effect, replied, “Instead of three circles in a plane, imagine three balls lying on a surface plate. Instead of drawing tangents, imagine a cone wrapped around each pair of balls. The apexes of the three cones will then lie on the surface plate. On top of the balls lay another surface plate. It will rest on the three balls and will be necessarily tangent to each of the three cones, and will contain the apexes of the three cones. Thus the apexes of the three cones will lie in both of the two plates, which is of course a straight line.//

Let P,Q,R be points on the xy plane, $\overrightarrow{p}$, $\overrightarrow{q}$, $\overrightarrow{r}$ be position vectors of P,Q,R respectively.

(a) Prove that

(i) if P,Q,R are collinear, and $x\overrightarrow{p}+y\overrightarrow{q}+z\overrightarrow{r}=\overrightarrow{0}$ for some scalars x,y and z, then $x+y+z=0$;

(solution)

(ii) if $x\overrightarrow{p}+y\overrightarrow{q}+z\overrightarrow{r}=\overrightarrow{0}$ for some scalars x,y and z, such that $x+y+z=0$ and not all x,y,z are zeros, then P,Q,R are collinear.

(solution)

(b) Let $O_1,O_2,O_3$ be circles centred at A,B,C respectively. Let $r_1,r_2,r_3$ be radii of $O_1,O_2,O_3$ respectively, where $r_1 > r_2 > r_3$. None of circle lies completely inside another. Let P,Q,R be the points of intersection of common external tangents of $O_1$ & $O_2$, $O_2$ & $O_3$, $O_3$ & $O_1$ respectively. Let $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}, \overrightarrow{p}, \overrightarrow{q}, \overrightarrow{r}$ be the position vectors of A,B,C,P,Q,R respectively.

(i) By considering similar triangles, or otherwise,
prove that $\overrightarrow{AP}=\frac{r_1}{r_1-r_2}\overrightarrow{AB}$, and hence $(r_1-r_2)\overrightarrow{p}=r_1\overrightarrow{b}-r_2\overrightarrow{a}$.

(solution)

(ii) Evaluate $r_3(r_1-r_2)\overrightarrow{p}+r_1(r_2-r_3)\overrightarrow{q}+r_2(r_3-r_1)\overrightarrow{r}$.

(solution)

(iii) Prove that P,Q,R are collinear.

(solution)

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*（附錄）我中學畢業於一所 band 5 的工業學校，以下作圖法應該在初中的幾何繪圖堂學過的。