MathDB

5

Part of 2021 Novosibirsk Oral Olympiad in Geometry

Problems(3)

<BCA=? if 2AP=BC, BX=BY, symmetrics (2021 Novosibirsk Oral Geo Oly 7.5)

Source:

4/27/2021
In an acute-angled triangle ABCABC on the side ACAC, point PP is chosen in such a way that 2AP=BC2AP = BC. Points XX and YY are symmetric to PP with respect to vertices AA and CC, respectively. It turned out that BX=BYBX = BY. Find ∠BCA\angle BCA.
geometryanglesequal segments
BP=PQ wanted, CD=CE, right isosceles (2021 Novosibirsk Oral Geo Oly 8.5)

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5/24/2021
On the legs ACAC and BCBC of an isosceles right-angled triangle with a right angle CC, points DD and EE are taken, respectively, so that CD=CECD = CE. Perpendiculars on line AEAE from points CC and DD intersect segment ABAB at points PP and QQ, respectively. Prove that BP=PQBP = PQ.
geometryright triangleisoscelesequal segments
<CNK=? <ECD=40^o, cyclic ABCDE (2021 Novosibirsk Oral Geo Oly 9.5)

Source:

5/25/2021
The pentagon ABCDEABCDE is inscribed in the circle. Line segments ACAC and BDBD intersect at point KK. Line segment CECE touches the circumcircle of triangle ABKABK at point NN. Find the angle CNKCNK if ∠ECD=40o.\angle ECD = 40^o.
geometrypentagonangles