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Miquel Triangle

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MiquelTriangle

A Miquel triangle of a reference triangle DeltaABC is a triangle DeltaP_AP_BP_C formed by choosing points P_A, P_B, and P_C on the side lines BC, CA, and AB, respectively. The three Miquel circles meet at a Miquel point M. A given M generally corresponds to more than one Miquel triangle.

Let P_A, P_B, and P_C lie fractional distances k_a, k_b, and k_c from B toward C, from C toward A, and from A toward B, 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_b^'k_(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_a^'k_(ab)^'-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.

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Miquel Triangle

Cite this as:

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

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