MathDB

2

Part of 2019 Novosibirsk Oral Olympiad in Geometry

Problems(3)

incircle of triangle inscribed in semicircle (2019 Novosibirsk Oral Geo Oly 8.2)

Source:

4/28/2021
The circle is inscribed in a triangle, inscribed in a semicircle. Find the marked angle aa. https://cdn.artofproblemsolving.com/attachments/8/e/334c8662377155086e9211da3589145f460b52.png
geometryincirclesemicircle
walks around or across an equilateral (2019 Novosibirsk Oral Geo Oly 7.2)

Source:

4/26/2021
Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of 600600 m, Krash and Wally live on the same shore, 300300 m from each other. In summer, Dokko to Krash walk 900900 m, and Wally to Rosa - also 900900 m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa.
[url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki
geometrydistanceEquilateral
circumcenter of ABC = incenter of ABD(2019 Novosibirsk Oral Geo Oly 9.2)

Source:

4/28/2021
An angle bisector ADAD was drawn in triangle ABCABC. It turned out that the center of the inscribed circle of triangle ABCABC coincides with the center of the inscribed circle of triangle ABDABD. Find the angles of the original triangle.
geometryincenter