perplexus.info :: Geometry : Locus in a triangle

Locus in a triangle (Posted on 2023-10-09)

Let M be a point in the triangle ABC such that

area(ABM)=2 * area(ACM)

Show that the locus of all such points is a straight line.

No Solution Yet Submitted by Danish Ahmed Khan

Rating: 4.0000 (1 votes)

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Solution

Comment 1 of 1

Let D be the point on BD=2DC. D is one possible location for M. Area(ABD)/area(ACD)=2

Now let M' be any point on line AD (except A)

area(ABM')/area(ABD) = AM'/AD = area(ACM')/area(ACD)

So area(ABM')/area(ACM')=2

If M' were not on line line AD the areas will change according to the ratio BD'/CD' where D' is the intersection of lines AM' and DC.

Finally a note: If we remove the restriction that M is inside triangle ABC, the locus includes a second line.

Posted by Jer on 2023-10-09 13:10:38

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