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D is the midpoint of the base of isosceles triangle ABC. Points G1 and G2 are the centroids of triangles ABD and triangle ACD, respectively. We know AD=8 and G1G2=4. What is the perimeter of triangle ABC?

Nov 28, 2018

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Since D is the midpoint of the base, it divides the isosceles triangle into two congruent triangles, ABD and ACD

Because of symmetry of these triangles, we can let G and G1 will lie at (-a, b) and (a, b)

And the midpoint of GG! will lie on AD.....call this point, L

Suppose that AF and AG are the medians that intersect the base BC at M and N

And these medians pass through G and G1, respectively

Because GG1 are the same distance from the base BC, they are parallel to BC

So....triangles ADN  and ALG1  are similar

This implies that  AG1/LG1 = AN/DN

LG1 is the midpoint of GG1   = 2

And, because AN, is a median...then AG1 = (2/3)AN

Substituting, we have

(2/3)AN / 2 = AN / DN  rearrange as

AN / (2/3)AN = DN /2

(3/2) = DN / 2

3 = DN

But this is (1/2) of DC...so DC = 6

And.....because of symmetry, DC = DB = 6

And because DN is perpendicular to BC.....then ADC is a right triangle

AD = 8, DC = 6......so.....AC is the hypotenuse = 10

And, by symmetry.....AC = AB = 10

So....the perimeter of ABC is

AC + AB + DB + DC =

10 + 10 + 6 + 6  =

32   Nov 30, 2018