MathDB

5

Part of 2024 India Regional Mathematical Olympiad

Problems(2)

Some Length Equality

Source: RMO 2024/5

11/3/2024
Let ABCDABCD be a cyclic quadrilateral such that AB∥CDAB \parallel CD. Let OO be the circumcenter of ABCDABCD and LL be the point on ADAD such that OLOL is perpendicular to ADAD. Prove that OB⋅(AB+CD)=OL⋅(AC+BD). OB\cdot(AB+CD) = OL\cdot(AC + BD).
geometrycyclic quadrilateral
RMO KV 2024 Q5

Source: RMO KV 2024 Q5

11/3/2024
Let ABCABC be a triangle with ∠ABC=20∘\angle ABC = 20^{\circ} and ∠ACB=40∘\angle ACB = 40^{\circ}. Let DD be a point on BCBC such that ∠BAD=∠DAC\angle BAD = \angle DAC. Let the incircle of triangle ABCABC touch BCBC at EE. Prove that BD=2⋅CEBD = 2 \cdot CE.
geometry