MathDB

3

Part of 2019 Regional Olympiad of Mexico Northwest

Problems(1)

min sum of areas of 2 circles, through, tangent to BC at B and C

Source: 2nd Mathematics Regional Olympiad of Mexico Northwest 2019 P3

9/7/2022
On a circle ω\omega with center O and radius rr three different points A,BA, B and CC are chosen. Let ω1\omega_1 and ω2\omega_2 be the circles that pass through AA and are tangent to line BCBC at points BB and CC, respectively. (a) Show that the product of the areas of ω1\omega_1 and ω2\omega_2 is independent of the choice of the points A,BA, B and CC. (b) Determine the minimum value that the sum of the areas of ω1\omega_1 and ω2\omega_2 can take and for what configurations of points A,BA, B and CC on ω\omega this minimum value is reached.
geometrycirclesgeometric inequality