MathDB

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

Source: Spanish Mathematical Olympiad 1990 P5

August 2, 2018

ratiogeometryarea of a triangleminimum

Problem Statement

On the sides

BC,CA

and

AB

of a triangle

ABC

of area

S

are taken points

A' ,B' ,C'

respectively such that

AC' /AB = BA' /BC = CB' /CA = p

, where

0 < p < 1

is variable. (a) Find the area of triangle

A' B' C'

in terms of

p

. (b) Find the value of

p

which minimizes this area. (c) Find the locus of the intersection point

P

of the lines through

A'

and

C'

parallel to

AB

and

AC

respectively.