AP is perpendicular to PC

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Source: China TST 2011 - Quiz 1 - D2 - P1

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1347 posts

WakeUp

#1 h May 19, 2011, 2:27 PM • 2 Y

Y by Adventure10, Mango247

Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$ . A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$ . Prove that $AP\perp PC$ .

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1347 posts

#2 h May 19, 2011, 4:27 PM • 1 Y

Y by Adventure10

It's somehow intresting to me that our teacher gave us this problem about $2$ years ago! for solving it, extend $PC$ to intersect the circle with center $O_1$ ...

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4147 posts

#3 h May 19, 2011, 4:39 PM • 1 Y

Y by Adventure10

Hmm, this is basically question 2 of 15 IBMO-Venezuela. See the following threads

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=46&t=83793
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=47&t=336494
http://www.artofproblemsolving.com/Forum/viewtopic.php?f=288&t=80971

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#4 h May 20, 2011, 2:27 AM • 5 Y

Y by ZacPower123, trying_to_solve_br, Adventure10, Mango247, and 1 other user

This solution assumes $AB$ is the closer tangent to $P$ , which I think the original problem states (it doesn't matter though).

Let $(O_1)\cap(O_2)=\{P,Q\}$ , $M=AB\cap PQ$ , and $A'$ be the reflection of $A$ over $O_1$ . Clearly $\angle{CPA}=90^\circ\Longleftrightarrow\angle{AA'C}=\angle{AA'P}$ . But $\angle{AA'P}=\angle{BAP}$ , so it suffices to show $\triangle{AA'C}\sim\triangle{BAP}\Longleftrightarrow\triangle{AO_1C}\sim\triangle{BMP}$ (equivalent since $O_1,M$ are the midpoints of $AA',BA$ respectively), which is just a simple angle chase.

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RaleD

#5 h May 28, 2011, 7:33 AM • 4 Y

Y by yttrerium, ZacPower123, Adventure10, Mango247

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#6 h Jul 2, 2012, 8:14 AM • 2 Y

Y by Adventure10, Mango247

RaleD wrote:

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This solution by RaleD is a very nice solution. At first glance, I thought it is similar to mine. In fact, it's way simpler and nicer. In my opinion, it's very clever to use the radical axis.

My solution also uses something like the equation $CN*CA=CP^2$ to get a geometric relation.

Here is how it goes:
Let the line through $C$ perpendicular to $AB$ intersect $AB$ at point $S$ .
A direct calculation of side lengths (starting with $BC^2-PC^2=AB^2-AP^2$ ) yields $AS\cdot AB=AP^2$
Now, it is easy to find that $A,S,C,P$ are concyclic, which leads to $\angle{APC}=90^\circ$

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#7 h Jan 10, 2015, 5:57 PM • 2 Y

Y by Adventure10, Mango247

Denote $Q$ the other intersection point of two circles,let $M$ be the intersection point of $PQ$ and $AB$ and let $O$ be the circumcenter of $ABQ$ .Now,let $C'$ be the intersection of the perpendicular from $A$ to $BP$ and the perpendicular from $P$ to $AP$ .Now,we have that $AOM$ is similar to $APC'$ so we get $APM$ is similar to $AC'P$ .Now,from an easy angle chase and using the previos similarity we get $QC'O$ is similar to $QPB$ and from this we get $QC'=PC'$ so $C'=C$ and that is it.

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695 posts

#8 h May 25, 2015, 6:34 PM • 2 Y

Y by Adventure10, Mango247

Here's another solution:

Define $Z \equiv AB \cap O_1O_2$ and let $ZP, AC$ cut $(O_1)$ for a second time at $Q, C'$ , respectively. Let $H$ be the projection of $A$ onto $O_1O_2.$ Note that $Z$ is the external center of homothety that maps $(O_2) \mapsto (O_1).$ It is clear that this homothety takes $B \mapsto A$ and $P \mapsto Q$ , so $AC \perp BP \implies AC \perp AQ.$ Hence, $\overline{QC'}$ is a diameter of $(O_1) \implies Q, C', O_1$ are collinear. Furthermore, note that $\angle ZHA = \angle ZAO_1 = 90^{\circ} \implies \triangle ZAH \sim \triangle ZO_1A \implies ZH \cdot ZO_1 = ZA^2 = ZP \cdot ZQ$ , where the last step follows from Power of a Point. Therefore, $H, O_1, P, Q$ are concyclic, so $\measuredangle PHC = \measuredangle PHO_1 = \measuredangle PQO_1 = \measuredangle PQC' = \measuredangle PAC' = \measuredangle PAC$ , where the angles are directed. Hence, $P, H, C, A$ are concyclic, so $\measuredangle APC = \measuredangle AHC = 90^{\circ}$ , as desired. $\square$

http://farm6.staticflickr.com/5469/18090868695_ce94a8844d_z.jpg

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2125 posts

#9 h Feb 15, 2020, 7:26 AM • 4 Y

Y by amar_04, DPS, GeoMetrix, Adventure10

Solution

This post has been edited 2 times. Last edited by AlastorMoody, Feb 16, 2020, 6:34 PM

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Arefe

#10 h Apr 15, 2020, 6:07 PM

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with using the special point on the median ( X ) , the problem sounds easy

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MagicalToaster53

159 posts

MagicalToaster53

#11 h Mar 17, 2024, 7:46 PM

Y by

My solution is pretty much the identical solution outlined through by @math154.

Let $P, Q$ be the points of intersection between circles with circumcenters $O_1, O_2$ with $P$ closer to line $AB$ than $Q$ . Let $X = PQ \cap O_1O_2$ , and $A_1, B_1$ denote the reflections of $A, B$ over $O_1, O_2$ , respectively. Also let $M$ denote the midpoint of line $AB$ . We then make the following claim:

Claim: $\triangle A_1O_1C \sim \triangle AMP$ .
Proof: We have that as $AA_1 \parallel BB_1$ , that $\measuredangle A_1OC = \measuredangle BO_2C = \measuredangle AMP,$ as $BMXO_2$ is also cyclic. $\square$

We now show that $A_1, C, P$ are collinear:

Claim: $A_1, C, P$ are collinear.
Proof: We wish to show $\measuredangle AA_1C = \measuredangle AA_1P$ . It then suffices to equivalently show $\triangle A_1O_1C \sim \triangle AMP$ . Indeed, we have that $\measuredangle AA_1C = \measuredangle MAP,$ and by the above claim/lemma, we obtain the desired similarlity. $\square$

Hence we have that $A_1, C, P$ are collinear, in particular $AP \perp PC$ , as desired. $\blacksquare$

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PROF65

#12 h Mar 26, 2024, 11:51 PM

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Let $T$ the symmetric of $A$ wrt $O_1O_2$ ;
$S$ be the ex-similicenter ; $Q$ the intersection of $(O_1)$ and $SP$ ;
$P',A'$ the antipodes of $P,A$ ;
$D$ the intersection of $A'P'$ and $O_1O_2$ ;
$C'$ the intersection of $PP'$ and $O_1O_2$ we will show that $C'=C$ :
by butterfly we deduce that $O_1C'=O_1D$ i.e. $PAP'A'$ is parallelogram.
More $A'(PQ,AT)=-1$ but $A'(C'D,O_1 T)=-1$ hence $A',D,Q$ are collinear.
Since $A'Q\parallel AC'$ then $AC'\perp AQ\parallel BP$
therefore $C'=C$ .

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#13 h Mar 27, 2024, 2:17 AM

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WakeUp wrote:

Let one of the intersection points of two circles with centres $O_1,O_2$ be $P$ . A common tangent touches the circles at $A,B$ respectively. Let the perpendicular from $A$ to the line $BP$ meet $O_1O_2$ at $C$ . Prove that $AP\perp PC$ .

By drawing the missing orthocentre in $\triangle APB$ , the problem becomes equivalent to a statement regarding the queue-config which is quite neat: In triangle $ABC$ with orthocentre $H$ and point $Q$ on arc $\widehat{BAC}$ such that $\angle AQH =90^{\circ}$ (hence $QH$ bisects $BC$ ), the line through $H$ parallel to $AB$ , the line $AC$ , and the perpendicular bisector of line $QH$ concur. Let $X, Y$ be points on $AB, AC$ such that $AXHY$ is a parallelogram. Then line $XY$ passes through the midpoint of $AH$ , i.e, the centre of the circumcircle of $\triangle AQH$ . Also, $(AX, AY, AH, AQ)=-1$ by projecting all on line $BC$ , hence $XY \parallel AQ$ or $XY \perp QH$ , hence $XY$ is the perpendicular bisector of $QH$ , and proving $Y$ lies on it was all we needed.

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