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10.6

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(2)

plane flew along a diamond-shaped route (II Soros Olympiad 1995-96 R1 10.6)

Source:

6/3/2024
The sports plane flew along a diamond-shaped route in windy weather. He flew through the first three sides of the rhombus in aa , bb and cc hours, respectively. How long did it take him to cover the fourth side of the diamond? (The speed of an aircraft is a vector equal to the sum of two vectors: the aircraft’s own speed and the wind speed. Wind speed is a constant vector. The aircraft’s own speed is a vector of constant length).
geometryalgebra
inradius wanted, 3 equal inradii related

Source: II Soros Olympiad 1995-96 R2 10-11.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/3/2024
On sides BCBC, CACA and ABAB of triangle ABCABC, points A1A_1, B1B_1, C1C_1 are taken, respectively, so that the radii of the circles inscribed in triangles A1BC1A_1BC_1, AB1C1AB_1C_1 and A1B1CA_1B_1C are equal to each other and equal to rr. The radius of the circle inscribed in triangle A1B1C1A_1B_1C_1 is equal to r1r_1. Find the radius of the circle inscribed in triangle ABCABC.
geometryinradiusequal circles