MathDB

4

Part of 2010 Dutch IMO TST

Problems(2)

cyclic ABCD with <ABD <DBC given, cyclic from 2 midpoints wanted

Source:

8/4/2019
Let ABCDABCD be a cyclic quadrilateral satisfying ∠ABD=∠DBC\angle ABD = \angle DBC. Let EE be the intersection of the diagonals ACAC and BDBD. Let MM be the midpoint of AEAE, and NN be the midpoint of DCDC. Show that MBCNMBCN is a cyclic quadrilateral.
geometrycyclic quadrilateralequal anglesmidpoint
square inscribed in circle, another tangent circle, equal segments wanted

Source:

8/4/2019
Let ABCDABCD be a square with circumcircle Γ1\Gamma_1. Let PP be a point on the arc ACAC that also contains BB. A circle Γ2\Gamma_2 touches Γ1\Gamma_1 in PP and also touches the diagonal ACAC in QQ. Let RR be a point on Γ2\Gamma_2 such that the line DRDR touches Γ2\Gamma_2. Proof that ∣DR∣=∣DA∣|DR| = |DA|.
geometrysquarecirclesequal segments