第49届IMO第一题是一道平面几何题:已知H是锐角三角形ABC的垂心,以边BC的中点为圆心,过点H的圆与直线BC相交于两点A_1,A_2;以边CA的中点为圆心,过点H的圆与直线CA相交于两点B_1,B_2;以边.AB的中点为圆心,过点H的圆与直线AB相交于两点C_1,C_2.证明:六点A_1,A_2,B_1,B_2,C_1,C_2共圆. The first question of the 49th IMO is a plane geometry problem. It is known that H is the center of inclination of the acute-angled triangle ABC, and the center of the edge BC is the center of the circle. The circle passing through the point H intersects the straight line BC at two points A_1, A_2; The center point of CA is the center of the circle. The circle passing through point H intersects with the line CA at two points B_1, B_2. The circle centered by the point .AB is the center point. The circle crossing point H intersects the line AB at two points C_1, C_2. : Six-point A_1, A_2, B_1, B_2, C_1, C_2 co-circle.