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Miquel Triangle


MiquelTriangle

Given a point P and a triangle DeltaABC, the Miquel triangle is the triangle DeltaP_AP_BP_C connecting the side points P_A, P_B, and P_C of DeltaABC with respect to which M is the Miquel point.

Let the points defining the Miquel circles be fractional distances k_a, k_b, and k_c along the sides BC, CA, and AB, respectively, and let k_i^'=1-k_i and k_(ij)^'=1-k_i-k_j. The Miquel triangle has side lengths

a^'=sqrt(a^2k_ck_b^'+b^2k_bk_(bc)+c^2k_ck_(bc))
(1)
b^'=sqrt(a^2k_ak_(ac)+b^2k_ak_c^'+c^2k_c^'k_(ac))
(2)
c^'=sqrt(a^2k_bk_a^'+b^2k_bk_(ab)^'+c^2k_bk_a^')
(3)

and area

 Delta_M=[1-(k_a+k_b+k_c)+k_ak_b+k_bk_c+k_ck_a]Delta,
(4)

where Delta is the area of the reference triangle.

In the special case k_a=k_b=k_c=1/2, the Miquel triangle becomes the medial triangle.

All Miquel triangles of a given point M are directly similar, and M is the similitude center in every case.


See also

Miquel Circles, Miquel Point, Miquel's Theorem

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References

Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., p. 81, 1995.Miquel, A. "Mémoire de Géométrie." Journal de mathématiques pures et appliquées de Liouville 1, 485-487, 1838.

Referenced on Wolfram|Alpha

Miquel Triangle

Cite this as:

Weisstein, Eric W. "Miquel Triangle." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MiquelTriangle.html

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