8

Part of 2008 Hanoi Open Mathematics Competitions

Problems(2)

shorter diagonal of a rhombus (2008 HOMC Junior Q8)

Source:

The sides of a rhombus have length

\Delta COD

and the area is

S

. What is the length of the shorter diagonal?

geometryrhombusdiagonal

Convex quadrilateral

Source: HOMC 2008

Consider a convex quadrilateral

ABCD

O

be the intersection of

AC

BD

M, N

be the centroid of

\Delta AOB

\Delta COD

P, Q

be orthocenter of

\Delta BOC

\Delta DOA

, respectively. Prove that

MN\bot PQ