Mass Point Geometry : Mastering the art

This article will take your through some of the beautiful geometry problems that will be solved using mass point geometry(MPG) and unveil its extensions & applications in the geometry domain.
You can read the article on MPG basics here : http://mbadecoded.blogspot.in/2015/03/mass-point-geometry-explored.html

So moving on to the agenda of this article, let's see our first problem :

Q1 In triangle ABC, medians AD and CE intersect at P, PE = 1.5 , PD = 2, and DE = 2.5. What is the area of AEDC?
a) 13  b) 13.5  c) 14  d) 14.5  e) 15
                             

Assign B mass m. Thus, because E is the midpoint of AB, A also has a mass of m.

Similarly, C has a mass of m. D and E each have a mass of 2m because they are between B and C and A and B respectively. Note that the mass of D is twice the mass of A, so AP must be twice as long as PD.

PD has length 2, so AP has length 4 and AD has length 6. Similarly, CP is twice PE and PE = 1.5, so CP = 3 and CE = 4.5.

Now note that triangle PED is a 3-4-5 right triangle with the right angle DPE. This means that the quadrilateral AEDC is a kite.

The area of a kite is half the product of the diagonals, AD and CE. Recall that they are 6 and 4.5 respectively, so the area of AEDC is 6*4.5/2 = 13.5


You can refer the alternate solutions to this problem here : http://artofproblemsolving.com/wiki/index.php/2013_AMC_10B_Problems/Problem_16


Q2 Triangle ABC has AB = 21, AC = 22 and BC = 20. Points D and E are located on AB and
AC, respectively, such that DE is parallel to BC and contains the centre of inscribed circle of triangle ABC. Then, DE = m/n, where m and n are relatively prime positive integers. Find m + n.

You can refer the alternate solutions to this problem here : http://artofproblemsolving.com/wiki/index.php/2001_AIME_I_Problems/Problem_7


Q3 In the given figure, DE || BC and AD : DB = 5 : 4 , find the ratio of areas of triangles BDE and BCF.
a) 70 : 81  b) 16 : 25  c) 56 : 81  d) 52 : 81

Let the area of triangle ABC be A. Then Area of triangle ADE = 5*5*A/[(5 + 4)(5 + 4)] = 25A/81
Applying MPG as following and getting the shown ratios.

We can say area of triangle BDE = (4/5)(25A/81) [Areas divided in the ratio of their shared base]
Then we find the area of CEF similarly and subtract these areas from total to get area of BCF and our answer.



Reference : You can try more interesting problems which haven't been discussed here in detail and try reaching the answer yourself using MPG.
http://artofproblemsolving.com/wiki/index.php/Mass_points




This brings us to the close of article series on Mass point geometry. I hope after going through the two articles here on blog and practicing problems on the given link you will start reaching a good confidence level while dealing with problems in this area.

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Cheers!
AS