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9.10

Part of IV Soros Olympiad 1997 - 98 (Russia)

Problems(2)

construct bisector using ruler (IV Soros Olympiad 1997-98 Correspondence 9.10)

Source:

6/1/2024
On the plane there is an image of a circle with a marked center. Let an arbitrary angle be drawn on this plane. Using one ruler, construct the bisector of this angle.
geometryangle bisectorgeometric constructionRuler
incenter lies on a chord of a circle passing through A and B

Source: IV Soros Olympiad 1997-98 R3 9.10 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/1/2024
A circle is drawn through vertices AA and BB of triangle ABCABC, intersecting sides ACAC and BCBC at points MM and PP. It is known that the segment MPMP contains the center of the circle inscribed in ABCABC. Find MPMP if AB=cAB = c, BC=aBC = a, CA=bCA=b.
geometryincenter