MathDB

2

Part of 1993 Iran MO (2nd round)

Problems(2)

existence of a point P [Iran Second Round 1993]

Source:

11/25/2010
Let ABCABC be an acute triangle with sides and area equal to a,b,ca, b, c and SS respectively. [color=#FF0000]Prove or disprove that a necessary and sufficient condition for existence of a point PP inside the triangle ABCABC such that the distance between PP and the vertices of ABCABC be equal to x,yx, y and zz respectively is that there be a triangle with sides a,y,za, y, z and area S1S_1, a triangle with sides b,z,xb, z, x and area S2S_2 and a triangle with sides c,x,yc, x, y and area S3S_3 where S1+S2+S3=S.S_1 + S_2 + S_3 = S.
geometrygeometry proposed
Infinitely many straight lines [Iran Second Round 1993]

Source:

11/25/2010
Show that if D1D_1 and D2D_2 are two skew lines, then there are infinitely many straight lines such that their points have equal distance from D1D_1 and D2.D_2.
geometry proposedgeometry