MathDB

1

Part of 2008 India Regional Mathematical Olympiad

Problems(2)

Segment equal to circumradius

Source: (Indian) RMO 2008 Problem 1

11/9/2008
Let ABC ABC be an acute angled triangle; let D,F D,F be the midpoints of BC,AB BC,AB respectively. Let the perpendicular from F F to AC AC and the perpendicular from B B ti BC BC meet in N N: Prove that ND ND is the circumradius of ABC ABC. [15 points out of 100 for the 6 problems]
geometrycircumcirclegeometric transformationreflectionrectangletrigonometrygeometry unsolved
Trig bash or angle chase

Source: An Excursion in Mathematics

6/25/2020
On a semicircle with diameter ABAB and centre SS, points CC and DD are given such that point CC belongs to arc ADAD. Suppose ∠CSD=120∘\angle CSD = 120^\circ. Let EE be the point of intersection of the straight lines ACAC and BDBD and FF the point of intersection of the straight lines ADAD and BCBC. Prove that EF=3ABEF=\sqrt{3}AB.
semicirclegeometryintersectionsstraight linestrigonometry