3

Part of 1983 Vietnam National Olympiad

Problems(2)

Locus of points whose area of pedal triangle is constant

Source: Vietnam MO 1983 P3

ABC

and a positive number

k

are given. Find the locus of a point

M

inside the triangle such that the projections of

M

on the sides of

\Delta ABC

form a triangle of area

k

geometrygeometry proposed

Tetrahedron with two opposite sides equal.

Source: Vietnam MO 1983 P6

Let be given a tetrahedron whose any two opposite edges are equal. A plane varies so that its intersection with the tetrahedron is a quadrilateral. Find the positions of the plane for which the perimeter of this quadrilateral is minimum, and find the locus of the centroid for those quadrilaterals with the minimum perimeter.

geometry3D geometrytetrahedronperimetergeometry unsolved