Slični zadaci

IMO Shortlist 1969 problem 16

$(CZS 5)$ A convex quadrilateral $ABCD$ with sides $AB = a, BC = b, CD = c, DA = d$ and angles $\alpha = \angle DAB, \beta = \angle ABC, \gamma = \angle BCD,$ and $\delta = \angle CDA$ is given. Let $s = \frac{a + b + c +d}{2}$ and $P$ be the area of the quadrilateral. Prove that $P^2 = (s - a)(s - b)(s - c)(s - d) - abcd \cos^2\frac{\alpha +\gamma}{2}$

Slični zadaci

Lista Tekst Dva stupca

Zadaci

IMO Shortlist 1997 problem 23

Let $ABCD$ be a convex quadrilateral. The diagonals $AC$ and $BD$ intersect at $K$ . Show that $ABCD$ is cyclic if and only if $AK \sin A + CK \sin C = BK \sin B + DK \sin D$ .

IMO Shortlist 1969 problem 70

$(YUG 2)$ A park has the shape of a convex pentagon of area $50000\sqrt{3} m^2$ . A man standing at an interior point $O$ of the park notices that he stands at a distance of at most $200 m$ from each vertex of the pentagon. Prove that he stands at a distance of at least $100 m$ from each side of the pentagon.