X₉₇₀

CENTER OF THE APOLLONIUS CIRCLE

The Apollonius circle is described at X₁₈₁ as the circle tangent to the three excircles and encompassing them. That X₉₇₀ is its center was noted on New Year’s Day, 2002, by Paul Yiu. (Hyacinthos #4619-4623)

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

X₉₇₀ (CENTER OF THE APOLLONIUS CIRCLE) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).

Constructed Elements

• Central Lines: {3, 6}, {5, 10}

• Related Centers: X₃ (CIRCUMCENTER), X₆ (SYMMEDIAN POINT), X₅ (NINE-POINT CENTER), X₁₀ (SPIEKER CENTER)

Trilinears: \(a[a^{3}(b + c)^{2} + a(ab + ac - 2bc)(b^{2} + c^{2}) - bc(b^{3} + c^{3}) - a(b^{4} + c^{4}) - (b^{5} + c^{5})] : :\)

Search Value (6, 9, 13): \(23.293029143569\)

▸ Alternate Coordinates1 form

• Trilinears: \((r^{2} - s^{2}) \cos A + 2rs \sin A\)

▸ Incident Central Lines6 lines

{1, 181}, {3, 6}, {5, 10}, {21, 51}, {40, 43}, {185, 411}

▸ Functional Relations5 relations

• X₉₇₀ = complement of X₁₀₄₄₁

• X₉₇₀ = {X₁₈₁,X₁₆₈₂}-harmonic conjugate of X₁

• X₉₇₀ = {X₃₇₁,X₃₇₂}-harmonic conjugate of X₅₀₁₉

• X₉₇₀ = inverse-in-excircles-radical-circle of X₃₈₁₄

• X₉₇₀ = perspector of Apollonius triangle and cross-triangle of ABC and Apollonius triangle

Notes

The Apollonius circle is the inverse-in-excircles-radical-circle of the nine-point circle. (Randy Hutson, December 10, 2016)