Difference between revisions of "2001 USAMO Problems/Problem 2"
Chezbgone2 (talk | contribs) m (→Solution 1: fixed asymptote) |
|||
Line 3: | Line 3: | ||
==Solution== | ==Solution== | ||
− | ==Solution 1== | + | ===Solution 1=== |
It is well known that the excircle opposite <math>A</math> is tangent to <math>\overline{BC}</math> at the point <math>D_2</math>. (Proof: let the points of tangency of the excircle with the lines <math>BC, AB, AC</math> be <math>D_3, F,G</math> respectively. Then <math>AB+BD_3=AB + BF=AF = AG = AC + AG=AC + CD_3</math>. It follows that <math>2CD_3 = AB + BC - AC</math>, and <math>CD_3 = s-b = BD_1 = CD_2</math>, so <math>D_3 \equiv D_2</math>.) | It is well known that the excircle opposite <math>A</math> is tangent to <math>\overline{BC}</math> at the point <math>D_2</math>. (Proof: let the points of tangency of the excircle with the lines <math>BC, AB, AC</math> be <math>D_3, F,G</math> respectively. Then <math>AB+BD_3=AB + BF=AF = AG = AC + AG=AC + CD_3</math>. It follows that <math>2CD_3 = AB + BC - AC</math>, and <math>CD_3 = s-b = BD_1 = CD_2</math>, so <math>D_3 \equiv D_2</math>.) | ||
Line 13: | Line 13: | ||
By Menelaus' Theorem on <math>\triangle ACD_2</math> with segment <math>\overline{BE_2}</math>, it follows that <math>\frac{CE_2}{E_2A} \cdot \frac{AP}{PD_2} \cdot \frac{BD_2}{BC} = 1 \Longrightarrow \frac{AP}{PD_2} = \frac{(c - (s-a)) \cdot a}{(a-(s-c)) \cdot AE_1} = \frac{a}{s-a}</math>. It easily follows that <math>AQ = D_2P</math>. <math>\blacksquare</math> | By Menelaus' Theorem on <math>\triangle ACD_2</math> with segment <math>\overline{BE_2}</math>, it follows that <math>\frac{CE_2}{E_2A} \cdot \frac{AP}{PD_2} \cdot \frac{BD_2}{BC} = 1 \Longrightarrow \frac{AP}{PD_2} = \frac{(c - (s-a)) \cdot a}{(a-(s-c)) \cdot AE_1} = \frac{a}{s-a}</math>. It easily follows that <math>AQ = D_2P</math>. <math>\blacksquare</math> | ||
− | ==Solution 2== | + | ===Solution 2=== |
The key observation is the following lemma. | The key observation is the following lemma. | ||
Line 31: | Line 31: | ||
Note: We used directed lengths in our calculations to avoid possible complications caused by the different shapes of triangle <math>ABC</math>. | Note: We used directed lengths in our calculations to avoid possible complications caused by the different shapes of triangle <math>ABC</math>. | ||
+ | |||
+ | === Solution 3 === | ||
+ | |||
+ | Here is a rather nice solution using barycentric coordinates: | ||
+ | |||
+ | Let <math>A</math> be <math>(1,0,0)</math>, <math>B</math> be <math>(0,1,0)</math>, and <math>C</math> be <math>(0,0,1)</math>. Let the side lengths of the triangle be <math>a,b,c</math> and the semi-perimeter <math>s</math>. | ||
+ | |||
+ | Now, <cmath>CD_1=s-c, BD_1=s-b, AE_1=s-a, CE_1=s-c.</cmath> Thus, <cmath>CD_2=s-b, BD_2=s-c, AE_2=s-c, CE_2=s-a.</cmath> | ||
+ | |||
+ | Therefore, <math>D_2=(0:s-b:s-c)</math> and <math>E_2=(s-a:0:s-c).</math> Clearly then, the non-normalized coordinates of <math>P=(s-a:s-b:s-c).</math> | ||
+ | |||
+ | Normalizing, we have that <cmath>D_2=\left(0,\frac{s-b}{a},\frac{s-c}{a}\right), E_2=\left(\frac{s-a}{b},0,\frac{s-c}{b}\right), P=\left(\frac{s-a}{s},\frac{s-b}{s},\frac{s-c}{s}\right).</cmath> | ||
+ | |||
+ | Now, we find the point <math>Q'</math> inside the triangle on the line <math>AD_2</math> such that <math>AQ'=D_2P</math>. It is then sufficient to show that this point lies on the incircle. | ||
+ | |||
+ | <math>P</math> is the fraction <math>\frac{s-a}{s}</math> of the way "up" the line segment from <math>D_2</math> to <math>A</math>. Thus, we are looking for the point that is <math>\frac{s-a}{s}</math> of the way "down" the line segment from <math>A</math> to <math>D_2</math>, or, the fraction <math>1-\frac{s-a}{s}</math> of the way "up". | ||
+ | |||
+ | Thus, <math>Q'</math> has normalized <math>x</math>-coordinate <math>1-\frac{s-a}{s}=\frac{a}{s}</math>. | ||
+ | |||
+ | As the line <math>AD_2</math> has equation <math>(s-c)y=(s-b)z</math>, it can easily be found that <math>Q'</math> lies at <math>\left(\frac{a^2}{as}, \frac{(s-a)(s-b)}{as}, \frac{(s-a)(s-c}{as}\right)</math>, or in non-normalized coordinates with sum <math>as</math>, at <math>(a^2,(s-a)(s-b),(s-a)(s-c)</math>. | ||
+ | |||
+ | Recalling that the equation of the incircle is <cmath>a^2yz+b^2xz+c^2xy+(x+y+z)[(s-a)^2x+(s-b)^2y+(s-c)^2z]=0.</cmath> We must show that this equation is true for <math>Q'</math>'s values of <math>x,y,z</math>. | ||
+ | |||
+ | Plugging in our values, this means showing that | ||
+ | <cmath>a^2(s-a)^2(s-b)(s-c)+a^2(s-a)[b^2(s-c)+c^2(s-b)]+as[a^2(s-a)^2+(s-a)(s-b)^3+(s-a)(s-c)^3]=0.</cmath> | ||
+ | Dividing by <math>a(s-a)</math>, this is just | ||
+ | <cmath>a(s-a)(s-b)(s-c)+a[b^2(s-c)+c^2(s-b)]+a^2s(s-a)+s(s-b)^3+s(s-c)^3=0.</cmath> | ||
+ | |||
+ | Plugging in the value of <math>s:</math> | ||
+ | <cmath>\frac{a(-a+b+c)(a-b+c)(a+b-c)}{8}+\frac{ab^2(a+b-c)}{2}+\frac{ac^2(a-b+c)}{2}+\frac{a^2(a+b+c)(-a+b+c)}{4}+\frac{(a+b+c)(a-b+c)^3}{16}+\frac{(a+b+c)(a+b-c)^3}{16}=0.</cmath> | ||
+ | <cmath>2a(-a+b+c)(a-b+c)(a+b-c)+8ab^2(a+b-c)+8ac^2(a-b+c)+4a^2(a+b+c)(-a+b+c)+(a+b+c)(a-b+c)^3+(a+b+c)(a+b-c)^3=0</cmath> | ||
+ | <cmath>2a[(-a+b+c)(a-b+c)(a+b-c)+4b^2(a+b-c)+4c^2(a-b+c)]+(a+b+c)[4a^2(-a+b+c)+(a-b+c)^3+(a+b-c)^3]=0</cmath> | ||
+ | The first bracket is just | ||
+ | <cmath>-a^3-b^3-c^3+a^2b+a^2c+b^2c+b^2a+c^2a+c^2b-2abc+4ab^2+4b^3-4b^2c+4ac^2-4bc^2+4c^3=-a^3+3b^3+3c^3+a^2b+a^2c-3b^2c+5b^2a+5c^2a-3c^2b-2abc</cmath> | ||
+ | and the second bracket is <cmath>-2a^3+4a^2b+4a^2c+6ab^2+6ac^2-12abc.</cmath> | ||
+ | Dividing everything by <math>2a</math> gives | ||
+ | <cmath>-a^3+3b^3+3c^3+a^2b+a^2c-3b^2c+5b^2a+5c^2a-3c^2b-2abc+(a+b+c)(-a^2+2ab+2ac+3b^2+3c^2-6bc),</cmath> which is <math>0</math>, as desired. | ||
+ | |||
+ | As <math>Q'</math> lies on the incircle and <math>AD_2</math>, <math>Q'=Q</math>, and our proof is complete. | ||
== See also == | == See also == |
Revision as of 20:33, 10 May 2016
Problem
Let be a triangle and let
be its incircle. Denote by
and
the points where
is tangent to sides
and
, respectively. Denote by
and
the points on sides
and
, respectively, such that
and
, and denote by
the point of intersection of segments
and
. Circle
intersects segment
at two points, the closer of which to the vertex
is denoted by
. Prove that
.
Solution
Solution 1
It is well known that the excircle opposite is tangent to
at the point
. (Proof: let the points of tangency of the excircle with the lines
be
respectively. Then
. It follows that
, and
, so
.)
Now consider the homothety that carries the incircle of to its excircle. The homothety also carries
to
(since
are collinear), and carries the tangency points
to
. It follows that
.
By Menelaus' Theorem on
with segment
, it follows that
. It easily follows that
.
Solution 2
The key observation is the following lemma.
Lemma: Segment is a diameter of circle
.
Proof: Let be the center of circle
, i.e.,
is the incenter of triangle
. Extend segment
through
to intersect circle
again at
, and extend segment
through
to intersect segment
at
. We show that
, which in turn implies that
, that is,
is a diameter of
.
Let be the line tangent to circle
at
, and let
intersect the segments
and
at
and
, respectively. Then
is an excircle of triangle
. Let
denote the dilation with its center at
and ratio
. Since
and
,
. Hence
. Thus
,
, and
. It also follows that an excircle
of triangle
is tangent to the side
at
.
It is well known that We compute
. Let
and
denote the points of tangency of circle
with rays
and
, respectively. Then by equal tangents,
,
, and
. Hence
It follows that
Combining these two equations yields
. Thus
that is,
, as desired.
Now we prove our main result. Let and
be the respective midpoints of segments
and
. Then
is also the midpoint of segment
, from which it follows that
is the midline of triangle
. Hence
and
. Similarly, we can prove that
.
2001usamo2-2.png
Let be the centroid of triangle
. Thus segments
and
intersect at
. Define transformation
as the dilation with its center at
and ratio
. Then
and
. Under the dilation, parallel lines go to parallel lines and the intersection of two lines goes to the intersection of their images. Since
and
,
maps lines
and
to lines
and
, respectively. It also follows that
and
or
This yields
as desired.
Note: We used directed lengths in our calculations to avoid possible complications caused by the different shapes of triangle .
Solution 3
Here is a rather nice solution using barycentric coordinates:
Let be
,
be
, and
be
. Let the side lengths of the triangle be
and the semi-perimeter
.
Now, Thus,
Therefore, and
Clearly then, the non-normalized coordinates of
Normalizing, we have that
Now, we find the point inside the triangle on the line
such that
. It is then sufficient to show that this point lies on the incircle.
is the fraction
of the way "up" the line segment from
to
. Thus, we are looking for the point that is
of the way "down" the line segment from
to
, or, the fraction
of the way "up".
Thus, has normalized
-coordinate
.
As the line has equation
, it can easily be found that
lies at
, or in non-normalized coordinates with sum
, at
.
Recalling that the equation of the incircle is We must show that this equation is true for
's values of
.
Plugging in our values, this means showing that
Dividing by
, this is just
Plugging in the value of
The first bracket is just
and the second bracket is
Dividing everything by
gives
which is
, as desired.
As lies on the incircle and
,
, and our proof is complete.
See also
2001 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.