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

Latest revision as of 04:48, 4 September 2024

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

Think about Radical Axis, Power of a Point and Radical Center.

Solution 1

Since $M$ is on the circle with diameter $AC$ , we have $\angle AMC=90$ and so $\angle MCA=90-A$ . We similarly 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 $PN\cdot PB=PM\cdot PC$ 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 name of the circle $AMND$ be $O$ . 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 $PT$ (a subsegment of $XY$ ) 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 $\angle AMC = \angle BND = \angle ATP = 90^\circ$ 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 $AM, XY$ , and $DN$ . This completes the problem.

Solution 3

Let $AM$ and $XY$ intersect at $Z$ . Because $\angle AMC = \angle BND = \angle APT = 90^\circ$ , we have quadrilaterals $AMPT$ and $DNPT$ cyclic. Therefore, $Z$ lies on the radical axis of the two circumcircles of these quadrilaterals. But $Z$ also lies on radical axis $XY$ of the original two circles, so the power of $Z$ with respect to each of the four circles is all equal to $ZM * ZA$ . Hence, $Z$ lies on the radical axis $DN$ of the two circles passing through $D$ and $N$ , as desired.

what is $T$ here?

Discussion

Lemma: The radical axis of two pairs of circles $O_1$ , $O_2$ and $O_3$ , $O_4$ are the same line $l$ . Furthermore, $O_1$ and $O_2$ intersect at $A$ and $B$ , and $O_3$ and $O_4$ intersect at $C$ and $D$ . Then $A, B, C,$ and $D$ are concyclic.

The proof of this lemma is trivial using the argument in Solution 3 and applying the converse of Power of a Point.

Note that this Problem 1 is a corollary of this lemma. This lemma is an effective way to relate four circles, just as the radical center can relate three circles.

Solution 1 also gives a trivial lemma that can also be useful:

Lemma 2: Chords $AB$ of $\omega_1$ and $CD$ of $\omega_2$ intersect on the segment $XY$ formed from the intersections of the two circles. Then $A, B, C, D$ are concyclic.

Two ways to solve a problem, two different insights into circle geometry. That is cool, but more RADICAL!

@@ Line 6: / Line 6: @@
 Think about Radical Axis, Power of a Point and Radical Center.
-== Solution ==
+== Solution 1==
 Since <math>M</math> is on the circle with diameter <math>AC</math>, we have <math>\angle AMC=90</math> and so <math>\angle MCA=90-A</math>.  We similarly find that <math>\angle BND=90</math>.  Also, notice that the line <math>XY</math> is the radical axis of the two circles with diameters <math>AC</math> and <math>BD</math>.  Thus, since <math>P</math> is on <math>XY</math>, we have <math>PN\cdot PB=PM\cdot PC</math> and so by the converse of Power of a Point, the quadrilateral <math>MNBC</math> is cyclic.  Thus, <math>90-A=\angle MCA=\angle BNM</math>.  Thus, <math>\angle MND=180-A</math> and so quadrilateral <math>AMND</math> is cyclic.  Let the name of the circle <math>AMND</math> be <math>O</math> .  Then, the radical axis of <math>O</math> and the circle with diameter <math>AC</math> is line <math>AM</math>.  Also, the radical axis of <math>O</math> and the circle with diameter <math>BD</math> is line <math>DN</math>.  Since the pairwise radical axes of 3 circles are concurrent, we have <math>AM,DN,XY</math> are concurrent as desired.
@@ Line 16: / Line 16: @@
 == Solution 3==
 Let <math>AM</math> and <math>XY</math> intersect at <math>Z</math>. Because <math>\angle AMC = \angle BND = \angle 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. But <math>Z</math> also lies on radical axis <math>XY</math> of the original two circles, so the power of <math>Z</math> with respect to each of the four circles is all equal to <math>ZM * ZA</math>. Hence, <math>Z</math> lies on the radical axis <math>DN</math> of the two circles passing through <math>D</math> and <math>N</math>, as desired.
+what is <math>T</math> here?
 ==Discussion==
@@ Line 33: / Line 35: @@
 [[Category:Olympiad Geometry Problems]]
+{{IMO box|year=1995|before=First Question|num-a=2}}

1995 IMO (Problems) • Resources
Preceded by First Question	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 2
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