MathDB

2

Part of 2021 Final Mathematical Cup

Problems(2)

D circumcenter of MNK wanted, < CAP = < CBP = <ACB

Source: 2021 3nd Final Mathematical Cup Junior Division P2 FMC

10/13/2021
Let ABCABC be an acute triangle, where ABAB is the smallest side and let DD be the midpoint of ABAB. Let PP be a point in the interior of the triangle ABCABC such that ∠CAP=∠CBP=∠ACB\angle CAP = \angle CBP = \angle ACB. From the point PP, we draw perpendicular lines on BCBC and ACAC where the intersection point with BCBC is MM, and with ACAC is NN . Through the point MM we draw a line parallel to ACAC, and through NN parallel to BCBC. These lines intercept at the point KK. Prove that DD is the center of the circumscribed circle for the triangle MNKMNK.
geometryCircumcenterequal angles
ZA tangent to (AXY), altitudes related

Source: 2021 3nd Final Mathematical Cup Senior Division P2 FMC

10/30/2022
The altitudes BB1BB_1 and CC1CC_1, are drawn in an acute triangle ABCABC. Let XX and YY be the points, which are symmetrical to the points B1B_1 and C1C_1, with respect to the midpoints of the sidesAB AB and ACAC of the triangle ABCABC respectively. Let's denote with ZZ the point of intersection of the lines BCBC and XYXY. Prove that the line ZAZA is tangent to the circumscribed circle of the triangle AXYAXY .
geometrytangent