MathDB

5

Part of 2024 Oral Moscow Geometry Olympiad

Problems(2)

Costruct symedian using only ruler

Source: Oral Moscow Geometry Olympiad 2024, 8-9.5

9/3/2024
An acute-angled unequal triangle ABCABC is drawn with its circumcircle and circumcenter OO. The incenter II is also marked. Using only a ruler (without divisions), construct the symedian (a line symmetrical to the median relative to the corresponding bisector) of the triangle, drawing no more than four lines.
geometrycircumcircleincenter
Locus of intersection of (A_1BC_2) and (A_2BC_1)

Source: Oral Moscow Geometry Olympiad 2024, 10-11.5

9/3/2024
From point DD of parallelogram ABCDABCD were drawn an arbitrary line ℓ1\ell_1, intersecting the segment ABAB and the line BCBC at points C1C_1 and A1A_1, respectively, and an arbitrary line ℓ2\ell_2 intersecting the segment BCBC and the line ABAB at the points A2A_2 and C2C_2, respectively. Find the locus of the intersection points of the circles (A1BC2)(A_1BC_2) and (A2BC1)(A_2BC_1) (other than point BB).
geometryparallelogram