MathDB

5

Part of 1990 Spain Mathematical Olympiad

Problems(1)

points defined by equal ratios on sides of triangle, min area of triangle, locus

Source: Spanish Mathematical Olympiad 1990 P5

8/2/2018
On the sides BC,CABC,CA and ABAB of a triangle ABCABC of area SS are taken points A′,B′,C′A' ,B' ,C' respectively such that AC′/AB=BA′/BC=CB′/CA=pAC' /AB = BA' /BC = CB' /CA = p, where 0<p<10 < p < 1 is variable. (a) Find the area of triangle A′B′C′A' B' C' in terms of p p. (b) Find the value of pp which minimizes this area. (c) Find the locus of the intersection point PP of the lines through A′A' and C′C' parallel to ABAB and ACAC respectively.
ratiogeometryarea of a triangleminimum