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9.7

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(2)

ratio chasing, AB/BC=AC/CD=l -- VI Soros Olympiad 1999-00 Round 1 9.7

Source:

5/21/2024
Points A,B,CA, B, C and DD are located on line ℓ\ell so that ABBC=ACCD=λ\frac{AB}{BC}=\frac{AC}{CD}=\lambda . A certain circle is tangent to line ℓ\ell at point CC. A line is drawn through AA that intersects this circle at points MM and NN such that the bisector perpendiculars to segments BMBM and DNDN intersect at point QQ on line ℓ\ell . In what ratio does point QQ divide segment ADAD?
ratiogeometry
equilateral wanted if 3 projections are equal

Source: VI Soros Olympiad 1990-00 R1 9.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/27/2024
In the acute-angled triangle ABCABC, the points PP, NN, M M are the feet of the altitudes drawn from the vertices CC, AA, BB, respectively. The lengths of the projections of the sides ABAB, BCBC, CACA on straight lines MNMN, PMPM, NPNP respectively, are equal to each other. Prove that triangle ABCABC is regular.
geometryEquilateral