MathDB

4

Part of 2022 Yasinsky Geometry Olympiad

Problems(4)

AK=BC if <AMB=60^o, BK=1/2 AC,

Source: 2022 Yasinsky Geometry Olympiad VIII-IX p4 / advanced p1 , Ukraine

11/7/2022
Let BMBM be the median of triangle ABCABC. On the extension of MBMB beyond BB, the point KK is chosen so that BK=12ACBK =\frac12 AC. Prove that if ∠AMB=60o\angle AMB=60^o, then AK=BCAK=BC.
(Mykhailo Standenko)
geometryequal segments
circumcircle of triangles by 3 angle bisectors, passes through A

Source: 2022 Yasinsky Geometry Olympiad VIII-IX advanced p4 , Ukraine

11/7/2022
Let XX be an arbitrary point on side BCBC of triangle ABCABC. Triangle TT is formed by the angle bisectors of the angles ∠ABC\angle ABC, ∠ACB\angle ACB and ∠AXC\angle AXC. Prove that the circle circumscribed around the triangle TT, passes through the vertex AA.
(Dmytro Prokopenko)
geometryAngle Bisectors
circumradii wanted, reflections of incenter wrt sides

Source: 2022 Yasinsky Geometry Olympiad X-XI p4 , Ukraine

11/8/2022
The intersection point II of the angles bisectors of the triangle ABCABC has reflections the points P,Q,TP,Q,T wrt the triangle's sides . It turned out that the circle ss circumscribed around of the triangle PQTPQT , passes through the vertex AA. Find the radius of the circumscribed circle of triangle ABCABC if BC=aBC = a.
(Gryhoriy Filippovskyi)
geometryincenter
collinearity wanted, AB+AC = 2BC, P is orthocenter of BIC

Source: 2022 Yasinsky Geometry Olympiad X-XI advanced p4 , Ukraine

11/8/2022
In the triangle ABCABC the relationship AB+AC=2BCAB+AC = 2BC holds. Let II and MM be the incenter and intersection point of the medians of triangle ABCABC respectively, ALAL its angle bisector, and point PP the orthocenter of triangle BICBIC. Prove that the points L,M,PL, M, P lie on a straight line.
(Matvii Kurskyi)
geometrycollinear