MathDB

2

Part of 2019 Tuymaada Olympiad

Problems(2)

prove that point lies on a common external tangent of 2 circles

Source: Tuymaada Olympiad 2019 junior p2

7/22/2019
A triangle ABCABC with AB<ACAB < AC is inscribed in a circle ω\omega. Circles γ1\gamma_1 and γ2\gamma_2 touch the lines ABAB and ACAC, and their centres lie on the circumference of ω\omega. Prove that CC lies on a common external tangent to γ1\gamma_1 and γ2\gamma_2.
geometrycircumcirclecommon tangentscircles
midpoint of segment circumcentres equidistant from 2 points

Source: Tuymaada 2019 p2

7/22/2019
A trapezoid ABCDABCD with BC//ADBC // AD is given. The points B′B' and C′C' are symmetrical to BB and CC with respect to CDCD and ABAB, respectively. Prove that the midpoint of the segment joining the circumcentres of ABC′ABC' and B′CDB'CD is equidistant from AA and DD.
geometrytrapezoidCircumcentermidpointdistance