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Show that the locus of the centroid G of triangle ABC is a circle that is traversed three times by G as A traversesC once, and determine the center and radius of this circle

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Problem 10542

(American Mathematical Monthly, Vol.103, August-September 1996) Proposed by J. Anglesio (France).

LetC be the circumcircle of a triangle A0B0C0 andI the incircle. It is known that, for each point A onC, there is a triangle ABC having C for circumcircle and I for incircle.

Show that the locus of the centroid G of triangle ABC is a circle that is traversed three times by G as A traversesC once, and determine the center and radius of this circle.

Solution proposed by Roberto Tauraso, Scuola Normale Superiore, piazza dei Cavalieri, 56100 Pisa, Italy.

Assume that the circumcircle C has center O = (0, 0) and radius R = 1 and the incircle I has center I= (d, 0) and radius r > 0.

Let ABC be a triangle having C for circumcircle and I for incircle. First, we will prove that its centroid G belongs to the circle CG centered in OG= (23d,0) and of radius rG= 13d2.

Let A = (x1, y1), B = (x2, y2), C = (x3, y3) then, since G = 13(A + B + C), we have to verify that (x1+ x2+ x3− 2d)2+ (y1+ y2+ y3)2= 9|GOG|2= d4.

We start by working out the left side of the equality

9|GOG|2= 9 − a2− b2− c2− 4d(x1+ x2+ x3) + 4d2, where a = |AB|, b = |BC| and c = |CA|.

By Ceva’s theorem [2], since I is the incentre of the triangle ABC, the following equalities hold

 ax1+ bx2+ cx3= d(a + b + c) ay1+ by2+ cy3= 0 .

If we sum the first equation multiplied by (x1+x2+x3) and the second one multiplied by (y1+y2+y3) we obtain that

d(x1+ x2+ x3) = 3 −a2(b + c) + b2(c + a) + c2(a + b) 2(a + b + c) . Therefore

9|GOG|2= −3 +−a3− b3− c3+ a2(b + c) + b2(c + a) + c2(a + b)

(a + b + c) + 4d2.

By the law of sines ([1] p.2) and Heron’s formula ([1] p.58), the area (ABC) of the triangle can be written in the following two ways

 8(ABC) = 2abc = 4r(a + b + c)

16(ABC)2= (a + b + c)(a + b − c)(a − b + c)(−a + b + c) = 4r2(a + b + c)2 . If we sum these two equations and divide the sum by (a + b + c), then

4r + 4r2= −a3− b3− c3+ a2(b + c) + b2(c + a) + c2(a + b)

(a + b + c) .

Therefore, since by Euler’s formula d2= 1 − 2r ([1] p.29),

9|GOG|2= −3 + 4r + 4r2+ 4d2= −3 + 2(1 − d2) + (1 − d2)2+ 4d2= d4.

As the point A rotates around C, the centroid G(A) of the triangle ABC having C for circumcircle and I for incircle moves continuously on CG(ABC exists by Poncelet’theorem).

Therefore, G( [P1P2) is a connected subset of CG and since

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G(P1) = G1= (13(2d + d2), 0), G(P2) = G2= (13(2d − d2), 0)

and G(·) is injective1 in the arc [P1P2, then G( [P1P2) is equal to the upper half of the circle CG. By simmetry

G( [P1P2) = −G( [P2P3) = G( [P3P4) = −G( [P4P5) = G( [P5P6) = −G( [P6P1), hence, as A traverses C once, CG is traversed exactly three times by G(A).

Note that, by a theorem due to Euler ([1] p.19), the circumcenter O the centroid G and the ortho- center H of the triangle ABC are collinear and OH = 3OG. Hence also the locus of orthocenter H

is a circle with center OH = 3OG and radius rH = 3rG. 

References

[1] H.S.M. Coxeter, S.L. Greitzer, Geometry Revisited, Math. Ass. of America, New York, 1967.

[2] S. Landy, A generalization of Ceva’s theorem to higher dimensions, Amer. Math. Monthly, 95 (1988), 936-939.

1If G ∈ CGthen G(P ) = G iff P is one of the three intersection points of the parametric curve x(t) = 3xG2(r + dcos t) cos t, y(t) = 3yG2(r + d cos t) sin t with t ∈ [0, 2π[ (cardioid) and the circle C.

Riferimenti

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