External Center of Similitude of Circles with respect to Radii
Theorem
Let $A$ and $B$ be the centers of two circles $\bigcirc Ar$ and $\bigcirc BR$ whose radii are $r$ and $R$ respectively, $r \ne R$.
Let $\bigcirc Ar$ and $\bigcirc BR$ be such that neither is completely enclosed inside the other.
Let $T$ be the external center of similitude of $\bigcirc Ar$ and $\bigcirc BR$.
Let $P$ and $Q$ be points on the circumference of $\bigcirc Ar$ and $\bigcirc BR$ respectively.
Then:
- $T$, $P$ and $Q$ are collinear.
Proof
Sufficient Condition
Let $AP$ and $QR$ are parallel.
We note that:
- $\angle TAP = \angle TBQ$
- $\angle PTA = \angle QTB$
This article, or a section of it, needs explaining. In particular: Why should $\angle PTA = \angle QTB$? Surely this is one of the things we have to prove? You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
and so by Triangles with Two Equal Angles are Similar:
- $\triangle TAP$ and $\triangle TBQ$ are similar.
Hence:
- $\dfrac {AT} {BT} = \dfrac r R$
As $P$ and $Q$ are arbitrary, it follows that the straight lines connecting the ends of all such parallel radii pass through $T$.
The same applies to the tangents, as they touch $\bigcirc Ar$ and $\bigcirc BR$ at parallel radii.
This article, or a section of it, needs explaining. In particular: The above also needs to be formally proved. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
$\Box$
Necessary Condition
Let $T$, $P$ and $Q$ be collinear.
Let $TP$ and $TQ$ cut $\bigcirc Ar$ and $\bigcirc BR$ at $P'$ and $Q'$.
Then $AP$ and $BQ$ are parallel as are $AP'$ and $BQ'$.
This article, or a section of it, needs explaining. In particular: This whole exposition of Clarke's is handwavey and unconvincing. Needs some proper Euclidean analysis on this. If nobody gets there before me, I'll go through and give it a think. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code. |
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {IV}$. Pure Geometry: Plane Geometry: The Centre of Similitude