MathDB

2

Part of 1999 Iran MO (2nd round)

Problems(2)

Making triangles - Iran NMO 1999 (Second Round) Problem2

Source:

10/4/2010
ABCABC is a triangle with ∠B>45∘\angle{B}>45^{\circ} , ∠C>45∘\angle{C}>45^{\circ}. We draw the isosceles triangles CAM,BANCAM,BAN on the sides AC,ABAC,AB and outside the triangle, respectively, such that ∠CAM=∠BAN=90∘\angle{CAM}=\angle{BAN}=90^{\circ}. And we draw isosceles triangle BPCBPC on the side BCBC and inside the triangle such that ∠BPC=90∘\angle{BPC}=90^{\circ}. Prove that ΔMPN\Delta{MPN} is an isosceles triangle, too, and ∠MPN=90∘\angle{MPN}=90^{\circ}.
symmetryvectorgeometry proposedgeometry
S_PQR/S_A'B'C' - Iran NMO 1999 (Second Round) Problem5

Source:

10/4/2010
Let ABCABC be a triangle and points P,Q,RP,Q,R be on the sides AB,BC,ACAB,BC,AC, respectively. Now, let A′,B′,C′A',B',C' be on the segments PR,QP,RQPR,QP,RQ in a way that AB∣∣A′B′AB||A'B' , BC∣∣B′C′BC||B'C' and AC∣∣A′C′AC||A'C'. Prove that: ABA′B′=SPQRSA′B′C′. \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}. Where SXYZS_{XYZ} is the surface of the triangle XYZXYZ.
geometry proposedgeometry