MathDB

2

Part of 2023 Yasinsky Geometry Olympiad

Problems(5)

equal interior and exterior angle bisectors

Source: 2023 Yasinsky Geometry Olympiad VIII-IX basic p2 , Ukraine

1/21/2024
In triangle ABCABC, the difference between angles BB and CC is equal to 90o90^o, and ALAL is the angle bisector of triangle ABCABC. The bisector of the exterior angle AA of the triangle ABCABC intersects the line BCBC at the point FF. Prove that AL=AFAL = AF.
(Alexander Dzyunyak)
geometryangle bisector
geo construction of incenter with min no of steps

Source: V.A. Yasinsky Geometry Olympiad 2023 VIII p2 , Ukraine

12/12/2023
Let II be the incenter of triangle ABCABC. K1K_1 and K2K_2 are the points on BCBC and ACAC respectively, at which the inscribed circle is tangent. Using a ruler and a compass, find the center of the inscribed circle for triangle CK1K2CK_1K_2 in the minimal possible number of steps (each step is to draw a circle or a line).
(Hryhorii Filippovskyi)
geometryincenterconstruction
CD= 2DE, (ABC) related

Source: V.A. Yasinsky Geometry Olympiad 2023 IX p2 , Ukraine

12/12/2023
Let BCBC and BDBD be the tangent lines to the circle with diameter ACAC. Let EE be the second point of intersection of line CDCD and the circumscribed circle of triangle ABCABC. Prove that CD=2DECD= 2DE.
(Matthew Kurskyi)
geometryequal segments
computational with bicentric ABCD, <ADB = 45^o

Source: 2023 Yasinsky Geometry Olympiad X-XI basic p2 , Ukraine

1/21/2024
Quadrilateral ABCDABCD is inscribed in a circle of radius RR, and also circumscribed around a circle of radius rr. It is known that ∠ADB=45o\angle ADB = 45^o. Find the area of triangle AIBAIB, where point II is the center of the circle inscribed in ABCDABCD.
(Hryhoriy Filippovskyi)
geometrybicentric quadrilateral
AZ _|_ BC if DX _|_ AB and DY _|_ AC, <BAC=60^o

Source: V.A. Yasinsky Geometry Olympiad 2023 X-XI p2 , Ukraine

12/13/2023
Let II be the center of the circle inscribed in triangle ABCABC which has ∠A=60o\angle A = 60^o and the inscribed circle is tangent to the sideBC at point DD. Choose points X andYon segments BIBI and CICI respectively, such than DX⊥ABDX \perp AB and DY⊥ACDY \perp AC. Choose a point ZZ such that the triangle XYZXYZ is equilateral and ZZ and II belong to the same half plane relative to the line XYXY. Prove that AZ⊥BCAZ \perp BC.
(Matthew Kurskyi)
geometryperpendicular