MathDB

2

Part of 2022 Durer Math Competition (First Round)

Problems(2)

R^2 = OP x OQ

Source: (2021 -) 2022 Dürer Math Competition Regional E2 https://artofproblemsolving.com/community/c1621671_

11/30/2022
In the acute triangle ABCABC the circle through BB touching the line ACAC at AA has centre PP, the circle through AA touching the line BCBC at BB has centre QQ. Let RR and OO be the circumradius and circumcentre of triangle ABCABC, respectively. Show that R2=OP⋅OQR^2 = OP \cdot OQ.
geometrycircumcircle
all triangles that can be split into 2 congruent pieces by 1 cut

Source: (2021 -) 2022 Dürer Math Competition Regional E+2 https://artofproblemsolving.com/community/c1621671_

11/30/2022
Determine all triangles that can be split into two congruent pieces by one cut. A cut consists of segments P1P2P_1P_2, P2P3P_2P_3, . . . , Pn−1PnP_{n-1}P_n where points P1,P2,...,PnP_1, P_2, . . . , P_n are distinct, points P1P_1 and PnP_n lie on the perimeter of the triangle and the rest of the points lie in the interior of the triangle such that the segments are disjoint except for the endpoints.
combinatorial geometrycongruent trianglesgeometry