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Greece Team Selection Test
2016 Greece Team Selection Test
2
2
Part of
2016 Greece Team Selection Test
Problems
(1)
Intersections of lines...
Source: 2016 Greece Team Selection Test,Problem 2
7/8/2016
Given is a triangle
△
A
B
C
\triangle{ABC}
△
A
BC
,with
A
B
<
A
C
<
B
C
AB<AC<BC
A
B
<
A
C
<
BC
,inscribed in circle
c
(
O
,
R
)
c(O,R)
c
(
O
,
R
)
.Let
D
,
E
,
Z
D,E,Z
D
,
E
,
Z
be the midpoints of
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
respectively,and
K
K
K
the foot of the altitude from
A
A
A
.At the exterior of
△
A
B
C
\triangle{ABC}
△
A
BC
and with the sides
A
B
,
A
C
AB,AC
A
B
,
A
C
as diameters,we construct the semicircles
c
1
,
c
2
c_1,c_2
c
1
,
c
2
respectively.Suppose that
P
≡
D
Z
∩
c
1
,
S
≡
K
Z
∩
c
1
P\equiv DZ\cap c_1 \ , \ S\equiv KZ\cap c_1
P
≡
D
Z
∩
c
1
,
S
≡
K
Z
∩
c
1
and
R
≡
D
E
∩
c
2
,
T
≡
K
E
∩
c
2
R\equiv DE\cap c_2 \ , \ T\equiv KE\cap c_2
R
≡
D
E
∩
c
2
,
T
≡
K
E
∩
c
2
.Finally,let
M
M
M
be the intersection of the lines
P
S
,
R
T
PS,RT
PS
,
RT
.i. Prove that the lines
P
R
,
S
T
PR,ST
PR
,
ST
intersect at
A
A
A
.ii. Prove that the lines
P
R
∩
M
D
PR\cap MD
PR
∩
M
D
intersect on
c
c
c
.[asy]import graph; size(8cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -4.8592569519241255, xmax = 12.331775417316715, ymin = -3.1864435704043403, ymax = 6.540061585876658; /* image dimensions */ pen aqaqaq = rgb(0.6274509803921569,0.6274509803921569,0.6274509803921569); pen uququq = rgb(0.25098039215686274,0.25098039215686274,0.25098039215686274); draw((0.6699432366054657,3.2576036755978928)--(0.,0.)--(5.,0.)--cycle, aqaqaq); /* draw figures */ draw((0.6699432366054657,3.2576036755978928)--(0.,0.), uququq); draw((0.,0.)--(5.,0.), uququq); draw((5.,0.)--(0.6699432366054657,3.2576036755978928), uququq); draw(shift((0.33497161830273287,1.6288018377989464))*xscale(1.662889476749906)*yscale(1.662889476749906)*arc((0,0),1,78.3788505217281,258.3788505217281)); draw(shift((2.834971618302733,1.6288018377989464))*xscale(2.7093067970187343)*yscale(2.7093067970187343)*arc((0,0),1,-36.95500560847834,143.0449943915217)); draw((0.6699432366054657,3.2576036755978928)--(0.6699432366054657,0.)); draw((-0.9938564482532047,2.628510486065423)--(2.5,0.)); draw((0.6699432366054657,0.)--(0.,3.2576036755978923)); draw((0.6699432366054657,0.)--(5.,3.257603675597893)); draw((2.5,0.)--(3.3807330143335355,4.282570444700163)); draw((-0.9938564482532047,2.628510486065423)--(2.5,4.8400585427926455)); draw((2.5,4.8400585427926455)--(5.,3.257603675597893)); draw((-0.9938564482532047,2.628510486065423)--(3.3807330143335355,4.282570444700163), linewidth(1.2) + linetype("2 2")); draw((0.,3.2576036755978923)--(5.,3.257603675597893), linewidth(1.2) + linetype("2 2")); draw(circle((2.5,1.18355242571055), 2.766007292905304), linewidth(0.4) + linetype("2 2")); draw((2.5,4.8400585427926455)--(2.5,0.), linewidth(1.2) + linetype("2 2")); /* dots and labels */ dot((0.6699432366054657,3.2576036755978928),linewidth(3.pt) + dotstyle); label("
A
A
A
", (0.7472169504504719,2.65), NE * labelscalefactor); dot((0.,0.),linewidth(3.pt) + dotstyle); label("
B
B
B
", (-0.2,-0.4), NE * labelscalefactor); dot((5.,0.),linewidth(3.pt) + dotstyle); label("
C
C
C
", (5.028818057451246,-0.34281415594345044), NE * labelscalefactor); dot((2.5,0.),linewidth(3.pt) + dotstyle); label("
D
D
D
", (2.4275434226319077,-0.32665717063401356), NE * labelscalefactor); dot((2.834971618302733,1.6288018377989464),linewidth(3.pt) + dotstyle); label("
E
E
E
", (3.073822835009383,1.5637101105701008), NE * labelscalefactor); dot((0.33497161830273287,1.6288018377989464),linewidth(3.pt) + dotstyle); label("
Z
Z
Z
", (0.003995626216375389,1.402140257475732), NE * labelscalefactor); dot((0.6699432366054657,0.),linewidth(3.pt) + dotstyle); label("
K
K
K
", (0.6179610679749769,-0.3105001853245767), NE * labelscalefactor); dot((-0.9938564482532047,2.628510486065423),linewidth(3.pt) + dotstyle); label("
P
P
P
", (-1.0785223895158957,2.7916409940873033), NE * labelscalefactor); dot((0.,3.2576036755978923),linewidth(3.pt) + dotstyle); label("
S
S
S
", (-0.14141724156855653,3.454077391774215), NE * labelscalefactor); dot((5.,3.257603675597893),linewidth(3.pt) + dotstyle); label("
T
T
T
", (5.061132028070119,3.3571354799175936), NE * labelscalefactor); dot((3.3807330143335355,4.282570444700163),linewidth(3.pt) + dotstyle); label("
R
R
R
", (3.445433497126431,4.375025554412117), NE * labelscalefactor); dot((2.5,4.8400585427926455),linewidth(3.pt) + dotstyle); label("
M
M
M
", (2.5567993051074027,4.940520040242407), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */[/asy]
geometry