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isosceles wanted, parallelogram and equal segments given, for juniors

Source: Dutch NMO 2014 p2 juniors

September 7, 2019
geometryparallelogramisoscelesIsosceles Triangleequal segments

Problem Statement

Let ABCDABCD be a parallelogram with an acute angle at AA. Let GG be a point on the line ABAB, distinct from BB, such that ∣CG∣=∣CB∣|CG| = |CB|. Let HH be a point on the line BCBC, distinct from BB, such that ∣AB∣=∣AH∣|AB| =|AH|. Prove that triangle DGHDGH is isosceles.
[asy] unitsize(1.5 cm);
pair A, B, C, D, G, H;
A = (0,0); B = (2,0); D = (0.5,1.5); C = B + D - A; G = reflect(A,B)*(C) + C - B; H = reflect(B,C)*(H) + A - B;
draw(H--A--D--C--G); draw(interp(A,G,-0.1)--interp(A,G,1.1)); draw(interp(C,H,-0.1)--interp(C,H,1.1)); draw(D--G--H--cycle, dashed);
dot("AA", A, SW); dot("BB", B, SE); dot("CC", C, E); dot("DD", D, NW); dot("GG", G, NE); dot("HH", H, SE); [/asy]
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