2
Part of 1994 Iran MO (2nd round)
Problems(2)
Find the angle alpha [Iran Second Round 1994]
Source:
11/26/2010
In the following diagram, is the center of the circle. If three angles and be equal, find
[asy]
unitsize(40);
import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffttww = rgb(1,0.2,0.4); pen qqwuqq = rgb(0,0.39,0);
draw(circle((0,0),2.33),ttttff+linewidth(2.8pt)); draw((-1.95,-1.27)--(0.64,2.24),ffttww+linewidth(2pt)); draw((0.64,2.24)--(1.67,-1.63),ffttww+linewidth(2pt)); draw((-1.95,-1.27)--(1.06,0.67),ffttww+linewidth(2pt)); draw((1.67,-1.63)--(-0.6,0.56),ffttww+linewidth(2pt)); draw((-0.6,0.56)--(1.06,0.67),ffttww+linewidth(2pt)); pair parametricplot0_cus(real t){
return (0.6*cos(t)+0.64,0.6*sin(t)+2.24);
}
draw(graph(parametricplot0_cus,-2.2073069497794027,-1.3111498158746024)--(0.64,2.24)--cycle,qqwuqq); pair parametricplot1_cus(real t){
return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);
}
draw(graph(parametricplot1_cus,0.06654165390165974,0.9342857038103908)--(-0.6,0.56)--cycle,qqwuqq); pair parametricplot2_cus(real t){
return (0.6*cos(t)+-0.6,0.6*sin(t)+0.56);
}
draw(graph(parametricplot2_cus,-0.766242589858673,0.06654165390165967)--(-0.6,0.56)--cycle,qqwuqq);
dot((0,0),ds); label("", (-0.2,-0.38), NE*lsf); dot((0.64,2.24),ds); label("", (0.72,2.36), NE*lsf); dot((-1.95,-1.27),ds); label("", (-2.2,-1.58), NE*lsf); dot((1.67,-1.63),ds); label("", (1.78,-1.96), NE*lsf); dot((1.06,0.67),ds); label("", (1.14,0.78), NE*lsf); dot((-0.6,0.56),ds); label("", (-0.92,0.7), NE*lsf); label("", (0.48,1.38),NE*lsf); label("", (-0.02,0.94),NE*lsf); label("", (0.04,0.22),NE*lsf); clip((-8.84,-9.24)--(-8.84,8)--(11.64,8)--(11.64,-9.24)--cycle);
[/asy]
geometry proposedgeometry
Prove that the points are collinear [Iran Second Round 1994]
Source:
11/26/2010
The incircle of triangle meet the sides and in and , respectively. Prove that the orthocenter of triangle the incenter and the circumcenter of triangle are collinear.[asy]
import graph; size(300); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen ttttff = rgb(0.2,0.2,1); pen ffwwww = rgb(1,0.4,0.4); pen xdxdff = rgb(0.49,0.49,1);
draw((8,17.58)--(2.84,9.26)--(20.44,9.21)--cycle); draw((8,17.58)--(2.84,9.26),ttttff+linewidth(2pt)); draw((2.84,9.26)--(20.44,9.21),ttttff+linewidth(2pt)); draw((20.44,9.21)--(8,17.58),ttttff+linewidth(2pt)); draw(circle((9.04,12.66),3.43),blue+linewidth(1.2pt)+linetype("8pt 8pt")); draw((6.04,14.42)--(8.94,9.24),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(11.12,15.48),ffwwww+linewidth(1.2pt)); draw((11.12,15.48)--(6.04,14.42),ffwwww+linewidth(1.2pt)); draw((8.94,9.24)--(7.81,14.79)); draw((11.12,15.48)--(6.95,12.79)); draw((6.04,14.42)--(10.12,12.6));
dot((8,17.58),ds); label("", (8.11,18.05),NE*lsf); dot((2.84,9.26),ds); label("", (2.11,8.85), NE*lsf); dot((20.44,9.21),ds); label("", (20.56,8.52), NE*lsf); dot((9.04,12.66),ds); label("", (8.94,12.13), NE*lsf); dot((6.04,14.42),ds); label("", (5.32,14.52), NE*lsf); dot((11.12,15.48),ds); label("", (11.4,15.9), NE*lsf); dot((8.94,9.24),ds); label("", (8.91,8.58), NE*lsf); dot((7.81,14.79),ds); label("", (7.81,15.14),NE*lsf); dot((6.95,12.79),ds); label("", (6.64,12.07),NE*lsf); dot((10.12,12.6),ds); label("", (10.41,12.35),NE*lsf); dot((8.07,13.52),ds); label("", (8.11,13.88),NE*lsf); clip((-0.68,-0.96)--(-0.68,25.47)--(30.71,25.47)--(30.71,-0.96)--cycle);
[/asy]
geometryincentercircumcircleinradiusgeometry proposed