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Difference between revisions of "1995 IMO Problems/Problem 1"

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==Solution 2==
 
==Solution 2==
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Let <math>AM</math> and <math>PT</math> intersect at <math>Z</math>. Now, assume that <math>Z, N, P</math> are not collinear. In that case, let <math>ZD</math> intersect the circle with diameter <math>BD</math> at <math>N'</math>.
+
Let <math>AM</math> and <math>PT</math> intersect at <math>Z</math>. Now, assume that <math>Z, N, P</math> are not collinear. In that case, let <math>ZD</math> intersect the circle with diameter <math>BD</math> at <math>N'</math> and the circle through <math>D, P, T</math> at <math>N''</math>.
  
−
We know that <math><AMC = <BND = <ATP = 90^\circ</math> via standard formulae, so quadrilaterals <math>AMPT</math> and <math>DNPT</math> are cyclic. Hence, by Power of a Point, <cmath>ZM * ZA = ZP * ZT = ZN * ZD.</cmath> However, because <math>Z</math> lies on radical axis <math>TP</math> of the two circles, we have <cmath>ZM * ZA = ZN' * ZD.</cmath> Hence, <math>ZD = ZD'</math>, a contradiction since <math>D</math> and <math>D'</math> are distinct. We therefore conclude that <math>Z, N, D</math> are collinear, which gives the concurrency of <math>AM, PT</math>, and <math>DN</math>. This completes the problem.
+
We know that <math><AMC = <BND = <ATP = 90^\circ</math> via standard formulae, so quadrilaterals <math>AMPT</math> and <math>DNPT</math> are cyclic. Thus, <math>N'</math> and <math>N''</math> are distinct, as none of them is <math>N</math>. Hence, by Power of a Point, <cmath>ZM * ZA = ZP * ZT = ZN'' * ZD.</cmath> However, because <math>Z</math> lies on radical axis <math>TP</math> of the two circles, we have <cmath>ZM * ZA = ZN' * ZD.</cmath> Hence, <math>ZN'' = ZN'</math>, a contradiction since <math>D</math> and <math>D'</math> are distinct. We therefore conclude that <math>Z, N, D</math> are collinear, which gives the concurrency of <math>AM, PT</math>, and <math>DN</math>. This completes the problem.
 +
 
 +
== Solution 3==
 +
Let <math>AM</math> and <math>DN</math> intersect at <math>Z</math>. Because <math><AMC = <BND = <APT = 90^\circ</math>, we have quadrilaterals <math>AMPT</math> and <math>DNPT</math> cyclic. Therefore, <math>Z</math> lies on the radical axis of the two circumcircles of these quadrilaterals, so <math>ZM * ZA = ZN * ZD</math>. But as a result <math>Z</math> lies on the radical axis of the two original circles (with diameters <math>AC</math> and <math>BD</math>), so <math>Z</math> lies on <math>TP</math> as well.
  
 
==See also==
 
==See also==
  
 
[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]

Revision as of 08:54, 28 August 2014

Problem

Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM,DN,XY$ are concurrent.


Hint

Radical axis and radical center! Think RADICAL-ly!


Solution

Since $M$ is on the circle with diameter $AC$, we have $\angle AMC=90$ and so $\angle MCA=90-A$. We simlarly find that $\angle BND=90$. Also, notice that the line $XY$ is the radical axis of the two circles with diameters $AC$ and $BD$. Thus, since $P$ is on $XY$, we have $PN\cdotPB=PM\cdot PC$ (Error compiling LaTeX. Unknown error_msg) and so by the converse of Power of a Point, the quadrilateral $MNBC$ is cyclic. Thus, $90-A=\angle MCA=\angle BNM$. Thus, $\angle MND=180-A$ and so quadrilateral $AMND$ is cyclic. Let the circle which contains the points $AMND$ be cirle $O$. Then, the radical axis of $O$ and the circle with diameter $AC$ is line $AM$. Also, the radical axis of $O$ and the circle with diameter $BD$ is line $DN$. Since the pairwise radical axes of 3 circles are concurrent, we have $AM,DN,XY$ are concurrent as desired.

Solution 2

Let $AM$ and $PT$ intersect at $Z$. Now, assume that $Z, N, P$ are not collinear. In that case, let $ZD$ intersect the circle with diameter $BD$ at $N'$ and the circle through $D, P, T$ at $N''$.

We know that $<AMC = <BND = <ATP = 90^\circ$ via standard formulae, so quadrilaterals $AMPT$ and $DNPT$ are cyclic. Thus, $N'$ and $N''$ are distinct, as none of them is $N$. Hence, by Power of a Point, \[ZM * ZA = ZP * ZT = ZN'' * ZD.\] However, because $Z$ lies on radical axis $TP$ of the two circles, we have \[ZM * ZA = ZN' * ZD.\] Hence, $ZN'' = ZN'$, a contradiction since $D$ and $D'$ are distinct. We therefore conclude that $Z, N, D$ are collinear, which gives the concurrency of $AM, PT$, and $DN$. This completes the problem.

Solution 3

Let $AM$ and $DN$ intersect at $Z$. Because $<AMC = <BND = <APT = 90^\circ$, we have quadrilaterals $AMPT$ and $DNPT$ cyclic. Therefore, $Z$ lies on the radical axis of the two circumcircles of these quadrilaterals, so $ZM * ZA = ZN * ZD$. But as a result $Z$ lies on the radical axis of the two original circles (with diameters $AC$ and $BD$), so $Z$ lies on $TP$ as well.

See also

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