X₆₂₀

MIDPOINT OF X₆₁₈ AND X₆₁₉

X₆₂₀ is the midpoint of X₆₁₈ and X₆₁₉.

5, 6, 7 3, 4, 5 Kepler Kimberling

X₆₂₀ (MIDPOINT OF X₆₁₈ AND X₆₁₉) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {618, 619}

• Related Centers: X₆₁₈ (COMPLEMENT OF X₁₃), X₆₁₉ (COMPLEMENT OF X₁₄)

Trilinears: \([4S_{A}a^{2} - (b^{4} + c^{4})]/a : :\)

Search Value (6, 9, 13): \(2.831708957779\)

▸ Linear Combinations1 combination

• X₆₂₀ = \(3\) X₂ + X₉₉

▸ Incident Central Lines7 lines

{2, 99}, {3, 114}, {30, 625}, {98, 631}, {141, 542}, {187, 325}, {230, 538}

▸ Functional Relations11 relations

• X₆₂₀ = center of the hyperbola H that is the locus of perspectors of circumconics centered at a point on line X₂X₆, which is the locus of the X₂-Ceva conjugate of P as P moves on line X₂X₆. Also, H is the Kiepert hyperbola of the medial triangle; H is tangent to line X₂X₆ at X₂, and H passes X₃, X₃₉ and X₁₁₄. H is also the bicevian conic of X₂ and X₉₉ (Randy Hutson, February 10, 2016)

• X₆₂₀ = midpoint of X_i and X_j for these (i,j): (3, 114), (99, 115), (187, 325), (618, 619)

• X₆₂₀ = complement of X₁₁₅

• X₆₂₀ = X₁₈₇-of-X₂-Brocard-triangle

• X₆₂₀ = X₂₃₀-of-1st-Brocard-triangle

• X₆₂₀ = isotomic conjugate of isogonal conjugate of X₂₀₉₇₆

• X₆₂₀ = polar conjugate of isogonal conjugate of X₂₂₀₈₅

• X₆₂₀ = centroid of X₂X₃X₁₁₄X₂₄₈₂

• X₆₂₀ = Kosnita(X₉₉,X₂) point

• X₆₂₀ = crosssum of intersections of 1st and 2nd Lemoine circles (and the circle {X₁₆₈₇,X₁₆₈₈,PU(1),PU(2)}})

• X₆₂₀ = QA-P29 center (Complement of QA-P₂ wrt the Diagonal triangle) of quadrangle ABCX₂ (see http://www.chrisvantienhoven.nl/quadrangle-objects/15-mathematics/quadrangle-objects/artikelen-qa/57-qa-p29.html)

Notes

Let S = X₉₉. Let A' be the centroid of the triangle BCS, and define B' and C' cyclically. Let D' be the centroid of ABC. The four centroids form a quadrilateral homothetic to the quadrilateral ABCS. The center of homothety is X₆₂₀, which is the centroid of ABCS. (Randy Hutson, 9/23/2011)

Let A'B'C' and A″B″C″ be the (equilateral) antipedal triangles of X₁₃ and X₁₄, resp. Let A* be the midpoint of A' and A″, and define B* and C* cyclically. The triangle A*B*C* is homothetic to ABC, and the center of homothety is X₆₂₀. (Randy Hutson, February 10, 2016)

X₆₂₀ is the center of the inellipse that is the barycentric square of line X₂X₆. The Brianchon point (perspector) of the inellipse is X₄₅₉₀. (Randy Hutson, October 15, 2018)