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Difference between revisions of "Sparrow’s lemmas"

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Lemma 2 is partial case of Lemma 3 with spiral center <math>I,</math> and equal speeds (from <math>D</math> to <math>A</math> and from <math>E</math> to <math>C</math>). Start positions of these points are <math>D</math> and <math>E.</math>
 
Lemma 2 is partial case of Lemma 3 with spiral center <math>I,</math> and equal speeds (from <math>D</math> to <math>A</math> and from <math>E</math> to <math>C</math>). Start positions of these points are <math>D</math> and <math>E.</math>
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==Russian Math Olympiad 2011==
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[[File:Sparrow problem 1.png|300px|right]]
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Let triangle <math>ABC</math> with circumcircle <math>\Omega</math> be given.
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Let <math>K \in \Omega</math> be the midpoint of the arc <math>BC</math> which contain the point <math>A, M</math> be the midpoint of <math>BC, I_B</math> be incenter of <math>\triangle ABM, I_C</math> be incenter of <math>\triangle ACM.</math>
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Prove that points <math>A, I_B, I_C,</math> and <math>K</math> are concyclic.
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<i><b>Proof</b></i>
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Denote <math>\omega = \odot AI_BI_C, D = \omega \cap AB,</math>
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<math>E = \omega \cap AC, F = \omega \cap AM.</math>
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We use Lemma 2 for <math>\triangle ABM</math> and get <math>BD + FM = BM.</math>
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We use Lemma 2 for <math>\triangle ACM</math> and get <math>CE + FM = CM = BM \implies BD = CE.</math>
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We use Lemma 1 for <math>\triangle ABC</math> and get result: points <math>A,D,E,</math> and <math>K</math> are concyclic <math>\blacksquare</math>

Revision as of 15:20, 17 September 2025

Sparrow’s lemmas have been known to Russian Olympiad participants since at least 2016. Page was made by vladimir.shelomovskii@gmail.com, vvsss

Sparrow's Lemma 1

Sparrow 1.png

Let triangle $ABC$ with circumcircle $\Omega$ and points $D$ and $E$ on the sides $AB$ and $AC,$ respectively be given.

Let $K \in \Omega$ be the midpoint of the arc $BC$ which contain the point $A.$

Prove that $BD = CE$ iff points $A, D, E,$ and $K$ are concyclic.

Proof

$BK = CK, \angle ABK = \angle ACK.$

Let $BD = CE \implies \triangle BKD = \triangle CKE \implies$

$\angle KDA = \angle KEA \implies A, D, E,$ and $K$ are concyclic.

Let $A, D, E,$ and $K$ are concyclic $\implies \angle KDA = \angle KEA \implies$

$\angle BKD = \angle CKE \implies \triangle BKD = \triangle CKE \implies BD = CE.$

Sparrow’s Lemma 2

Sparrow 1A.png

Let triangle $ABC$ with circumcircle $\Omega$ and points $D$ and $E$ on the sides $AB$ and $AC,$ respectively be given.

Let $I$ be the incenter.

Prove that $BD + CE = BC$ iff points $A, D, E,$ and $I$ are concyclic.

Proof

1. Let points $A, D, E,$ and $I$ are concyclic.

Denote $F \in BC$ such $BD = BF, \varphi = \angle ADI.$

So point $F$ is symmetric to $D$ with respect to $BI \implies \angle BDI = 180^\circ - \varphi, \angle IEC = \varphi.$ \[\triangle BDI = \triangle BFI \implies DI = FI, \angle CFI = \varphi.\] \[\angle DAI = \angle EAI \implies DI = EI = FI.\] \[\triangle CIF = \triangle CIE \implies CE = CF \implies BD + CE = BC \blacksquare\] 2. Let $BD + CE = BC \implies$ there is point $F$ such that $BF = BD, CF = CE \implies$ \[\triangle CIF = \triangle CIE, \triangle BIF = \triangle BID \implies\] \[180^\circ = \angle BFI + \angle CFI = \angle BDI + \angle CEI = \angle ADI + \angle AEI \blacksquare\]

Sparrow’s Lemma 3

Sparrow 3.png

Let lines $\ell$ and $\ell'$ and points $A_0 \in \ell$ and $B_0 \in \ell'$ be given, $O = \ell \cap \ell'.$

Points $A$ and $B$ moves along $\ell$ and $\ell',$ respectively with fixed speeds. At moment $t = 0,$ $A = A_0, B = O$, at moment $t_0$ $A = O, B = B_0.$

Prove that circle $\Omega = \odot OAB$ contain fixed point ($P$).

Proof

Let $\omega$ be the circle contains $O$ and $B_0$ and tangent to $\ell.$ Let $\omega'$ be the circle contains $O$ and $A_0$ and tangent to $\ell'.$ \[P = \omega \cap \omega' \ne O.\] It is known that $P$ is the spiral center of spiral similarity $T$ mapping segment $A_0O$ to $OB_0.$ The ratio of the speeds of points $A$ and $B$ is $\frac{AA_0}{BO} = \frac{OA_0}{B_0O},$ so $T$ mapping segment $AA_0$ to $BO.$ Therefore $\Omega$ contain the spiral center $P \blacksquare$

Corollary 1

Lemma 1 is partial case of Lemma 3 with spiral center $K,$ equal speeds and two positions of the pare moving points - $D,E$ and $B,C.$

Corollary 2

Lemma 2 is partial case of Lemma 3 with spiral center $I,$ and equal speeds (from $D$ to $A$ and from $E$ to $C$). Start positions of these points are $D$ and $E.$

Russian Math Olympiad 2011

Sparrow problem 1.png

Let triangle $ABC$ with circumcircle $\Omega$ be given. Let $K \in \Omega$ be the midpoint of the arc $BC$ which contain the point $A, M$ be the midpoint of $BC, I_B$ be incenter of $\triangle ABM, I_C$ be incenter of $\triangle ACM.$

Prove that points $A, I_B, I_C,$ and $K$ are concyclic.

Proof

Denote $\omega = \odot AI_BI_C, D = \omega \cap AB,$ $E = \omega \cap AC, F = \omega \cap AM.$

We use Lemma 2 for $\triangle ABM$ and get $BD + FM = BM.$

We use Lemma 2 for $\triangle ACM$ and get $CE + FM = CM = BM \implies BD = CE.$

We use Lemma 1 for $\triangle ABC$ and get result: points $A,D,E,$ and $K$ are concyclic $\blacksquare$

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