EM_|_AB and similar triangles wanted, <ACB = 90^o
Source: Dutch NMO 2021 p4
December 28, 2021
geometrysimilar trianglesperpendicularright triangle
Problem Statement
In triangle we have . The point is the midpoint of . The line through parallel to intersects in . The midpoint of line segment is . The lines and are perpendicular.
(a) Prove that triangles and are similar.
(b) Prove that and are perpendicular.[asy]
unitsize(1 cm);pair A, B, C, D, E, M;A = (0,0);
B = (4,0);
C = (2.6,2);
M = (A + B)/2;
D = (A + C)/2;
E = (C + D)/2;draw(A--B--C--cycle);
draw(C--M--D--B);dot("", A, SW);
dot("", B, SE);
dot("", C, N);
dot("", D, NW);
dot("", E, NW);
dot("", M, S);
[/asy]Be aware: the figure is not drawn to scale.