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X₂₅₂₁–X₂₅₉₃¶

Circle Similicenters, Radical Centers & Isoconjugates II

Similicenters and intersections involving the orthocentroidal, 2nd Brocard, Parry, and Gallatly circles (X₂₅₂₁–X₂₅₇₃), followed by isogonal and X_k-isoconjugates (X₂₅₇₄–X₂₅₉₃).

X₂₅₂₁–X₂₅₉₃ (Circle Similicenters, Radical Centers & Isoconjugates II) — 16 representative centers (from 73 in collection) constructed on acute reference triangle \((a, b, c) = (5, 6, 7)\) with unit baseline AB = \(1\).¶

X_2521 — RADICAL CENTER OF {BROCARD, BEVAN,MOSES} CIRCLES

X₂₅₂₁RADICAL CENTER OF {BROCARD, BEVAN,MOSES} CIRCLES

X_2522 — RADICAL CENTER OF {NINE-POINT,BROCARD, APOLLONIUS} CIRCLES

X₂₅₂₂RADICAL CENTER OF {NINE-POINT,BROCARD, APOLLONIUS} CIRCLES

X₂₅₂₂ = crossdifference of every pair of points on line X₁X₂₅

X_2523 — RADICAL CENTER OF {NINE-POINT,BROCARD, BEVAN} CIRCLES

X₂₅₂₃RADICAL CENTER OF {NINE-POINT,BROCARD, BEVAN} CIRCLES

X_2524 — RADICAL CENTER OF {NINE-POINT, BROCARD, 2nd BROCARD} CIRCLES

X₂₅₂₄RADICAL CENTER OF {NINE-POINT, BROCARD, 2nd BROCARD} CIRCLES

X₂₅₂₄ = reflection of X₂₄₈₉ in X₂₅₀₇

X_2525 — RADICAL CENTER OF {NINE-POINT, BROCARD, (X_4,2R)} CIRCLES

X₂₅₂₅RADICAL CENTER OF {NINE-POINT, BROCARD, (X₄,2R)} CIRCLES

X₂₅₂₅ = X₃₀₅-Ceva conjugate of X₁₂₅

X_2526 — RADICAL CENTER OF {NINE-POINT, APOLLONIUS, (X_4,2R)} CIRCLES

X₂₅₂₆RADICAL CENTER OF {NINE-POINT, APOLLONIUS, (X₄,2R)} CIRCLES

X₂₅₂₆ = reflection of X₆₅₀ in X₁₄₉₁

X_2527 — RADICAL CENTER OF {NINE-POINT, SPIEKER, BEVAN} CIRCLES

X₂₅₂₇RADICAL CENTER OF {NINE-POINT, SPIEKER, BEVAN} CIRCLES

X₂₅₂₇ = midpoint of X₂₅₁₆ and X₂₅₂₉

X_2528 — RADICAL CENTER OF {ORTHOCENTROIDAL, BROCARD, (X_4,2R)} CIRCLES

X₂₅₂₈RADICAL CENTER OF {ORTHOCENTROIDAL, BROCARD, (X₄,2R)} CIRCLES

X₂₅₂₈ = X₉₉-Ceva conjugate of X₁₄₁

X_2529 — RADICAL CENTER OF {ORTHOCENTROIDAL, SPIEKER, BEVAN} CIRCLES

X₂₅₂₉RADICAL CENTER OF {ORTHOCENTROIDAL, SPIEKER, BEVAN} CIRCLES

X₂₅₂₉ = reflection of X₂₅₁₆ and X₂₅₃₇

X_2530 — RADICAL CENTER OF {BROCARD, BEVAN, (X_4,2R)} CIRCLES

X₂₅₃₀RADICAL CENTER OF {BROCARD, BEVAN, (X₄,2R)} CIRCLES

X₂₅₃₀ = reflection of X₆₆₇ and X₉₀₅

X_2531 — RADICAL CENTER OF {BROCARD, (X_4,2R), 2nd BROCARD} CIRCLES

X₂₅₃₁RADICAL CENTER OF {BROCARD, (X₄,2R), 2nd BROCARD} CIRCLES

X₂₅₃₁ = bicentric difference of PU(156)

X_2532 — RADICAL CENTER OF {APOLLONIUS, SPIEKER, BEVAN} CIRCLES

X₂₅₃₂RADICAL CENTER OF {APOLLONIUS, SPIEKER, BEVAN} CIRCLES

X_2533 — RADICAL CENTER OF {BEVAN, FUHRMANN, (X_4,2R)} CIRCLES

X₂₅₃₃RADICAL CENTER OF {BEVAN, FUHRMANN, (X₄,2R)} CIRCLES

X₂₅₃₃ = crosspoint of X₁₀ and X₆₆₈

X_2534 — INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₄INSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE)

The Spieker radical circle, with center X₁₀ and radius d/2, is introduced in

X_2535 — EXSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₅EXSIMILICENTER(INCIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₅ is the exsimilicenter(incircle, Spieker radical circle).

X_2536 — INSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₆INSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₆ is the insimilicenter(circumcircle, Spieker radical circle).

X_2537 — EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₇EXSIMILICENTER(CIRCUMCIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₇ is the exsimilicenter(circumcircle, Spieker radical circle).

X_2538 — INSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₈INSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₈ is the insimilicenter(Apollonius circle, Spieker radical circle).

X_2539 — EXSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₉EXSIMILICENTER(APOLLONIUS CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₃₉ is the exsimilicenter(Apollonius circle, Spieker radical circle).

X_2540 — INSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₄₀INSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₄₀ is the insimilicenter(nine-point circle, Spieker radical circle).

X_2541 — EXSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₄₁EXSIMILICENTER(NINE-POINT CIRCLE, SPIEKER RADICAL CIRCLE)

X₂₅₄₁ is the exsimilicenter(nine-point circle, Spieker radical circle).

X_2542 — INSIMILICENTER(BROCARD CIRCLE, (X_4,2R))

X₂₅₄₂INSIMILICENTER(BROCARD CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2543 — EXSIMILICENTER(BROCARD CIRCLE, (X_4,2R))

X₂₅₄₃EXSIMILICENTER(BROCARD CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2544 — INSIMILICENTER(GALLATLY CIRCLE, (X_4,2R))

X₂₅₄₄INSIMILICENTER(GALLATLY CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2545 — EXSIMILICENTER(GALLATLY CIRCLE, (X_4,2R))

X₂₅₄₅EXSIMILICENTER(GALLATLY CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2546 — INSIMILICENTER(1st LEMOINE CIRCLE, (X_4,2R))

X₂₅₄₆INSIMILICENTER(1st LEMOINE CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2547 — EXSIMILICENTER(1st LEMOINE CIRCLE, (X_4,2R))

X₂₅₄₇EXSIMILICENTER(1st LEMOINE CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2548 — INSIMILICENTER(MOSES CIRCLE, (X_4,2R))

X₂₅₄₈INSIMILICENTER(MOSES CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2549 — EXSIMILICENTER(MOSES CIRCLE, (X_4,2R))

X₂₅₄₉EXSIMILICENTER(MOSES CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2550 — INSIMILICENTER(SPIEKER CIRCLE, (X_4,2R))

X₂₅₅₀INSIMILICENTER(SPIEKER CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2551 — EXSIMILICENTER(SPIEKER CIRCLE, (X_4,2R))

X₂₅₅₁EXSIMILICENTER(SPIEKER CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2552 — INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X_4,2R))

X₂₅₅₂INSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2553 — EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X_4,2R))

X₂₅₅₃EXSIMILICENTER(ORTHOCENTROIDAL CIRCLE, (X₄,2R))

The circle (X₄,2R) is identified at X₂₄₇₄.

X_2554 — 1st INTERSECTION(EULER LINE, 2nd BROCARD CIRCLE)

X₂₅₅₄1st INTERSECTION(EULER LINE, 2nd BROCARD CIRCLE)

Of the two points of intersection, X₂₅₅₄ is one closer to X₄.

X_2555 — 2nd INTERSECTION(EULER LINE, 2nd BROCARD CIRCLE)

X₂₅₅₅2nd INTERSECTION(EULER LINE, 2nd BROCARD CIRCLE)

Of the two points of intersection, X₂₅₅₅ is one farther from X₄.

X_2556 — 1st INTERSECTION(LINE X_1X_3, 2nd BROCARD CIRCLE)

X₂₅₅₆1st INTERSECTION(LINE X₁X_3, 2nd BROCARD CIRCLE)

Of the two points of intersection, X₂₅₅₆ is one closer to X₁.

X_2557 — 2nd INTERSECTION(LINE X_1X_3, 2nd BROCARD CIRCLE)

X₂₅₅₇2nd INTERSECTION(LINE X₁X_3, 2nd BROCARD CIRCLE)

Of the two points of intersection, X₂₅₅₇ is one farther from X₁.

X_2558 — INSIMILICENTER(1st LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₅₈INSIMILICENTER(1st LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₅₈ is the insimilicenter(1st Lemoine circle, 2nd Brocard circle).

X_2559 — EXSIMILICENTER(1st LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₅₉EXSIMILICENTER(1st LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₅₉ is the exsimilicenter(1st Lemoine circle, 2nd Brocard circle).

X_2560 — INSIMILICENTER(2nd LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₀INSIMILICENTER(2nd LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₀ is the insimilicenter(2nd Lemoine circle, 2nd Brocard circle).

X_2561 — EXSIMILICENTER(2nd LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₁EXSIMILICENTER(2nd LEMOINE CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₁ is the exsimilicenter(2nd Lemoine circle, 2nd Brocard circle).

X_2562 — INSIMILICENTER(GALLATLY CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₂INSIMILICENTER(GALLATLY CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₂ is the insimilicenter(gallatly circle, 2nd Brocard circle).

X_2563 — EXSIMILICENTER(GALLATLY CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₃EXSIMILICENTER(GALLATLY CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₃ is the exsimilicenter(gallatly circle, 2nd Brocard circle).

X_2564 — INSIMILICENTER(INCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₄INSIMILICENTER(INCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₄ is the insimilicenter(incircle, 2nd Brocard circle).

X_2565 — EXSIMILICENTER(INCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₅EXSIMILICENTER(INCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₅ is the exsimilicenter(incircle, 2nd Brocard circle).

X_2566 — INSIMILICENTER(NINE-POINT CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₆INSIMILICENTER(NINE-POINT CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₆ is the insimilicenter(nine-point circle, 2nd Brocard circle).

X_2567 — EXSIMILICENTER(NINE-POINT CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₇EXSIMILICENTER(NINE-POINT CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₇ is the exsimilicenter(nine-point circle, 2nd Brocard circle).

X_2568 — INSIMILICENTER(SPIEKER CIRCLE, 2ndBROCARD CIRCLE)

X₂₅₆₈INSIMILICENTER(SPIEKER CIRCLE, 2ndBROCARD CIRCLE)

X₂₅₆₈ is the insimilicenter(Spieker circle, 2ndbrocard circle).

X_2569 — EXSIMILICENTER(SPIEKER CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₉EXSIMILICENTER(SPIEKER CIRCLE, 2nd BROCARD CIRCLE)

X₂₅₆₉ is the exsimilicenter(Spieker circle, 2nd Brocard circle).

X_2570 — INSIMILICENTER(ORTHOCENTROIDALCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₇₀INSIMILICENTER(ORTHOCENTROIDALCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₇₀ is the insimilicenter(orthocentroidalcircle, 2nd Brocard circle).

X_2571 — EXSIMILICENTER(ORTHOCENTROIDALCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₇₁EXSIMILICENTER(ORTHOCENTROIDALCIRCLE, 2nd BROCARD CIRCLE)

X₂₅₇₁ is the exsimilicenter(orthocentroidalcircle, 2nd Brocard circle).

X_2572 — INSIMILICENTER(BEVAN, 2nd BROCARD CIRCLE)

X₂₅₇₂INSIMILICENTER(BEVAN, 2nd BROCARD CIRCLE)

X₂₅₇₂ is the insimilicenter(Bevan, 2nd Brocard circle).

X_2573 — EXSIMILICENTER(BEVAN, 2nd BROCARD CIRCLE)

X₂₅₇₃EXSIMILICENTER(BEVAN, 2nd BROCARD CIRCLE)

X₂₅₇₃ is the exsimilicenter(Bevan, 2nd Brocard circle).

X_2574 — ISOGONAL CONJUGATE OF X_1113

X₂₅₇₄ISOGONAL CONJUGATE OF X₁₁₁₃

As the isogonal conjugate of a point on the circumcircle, X₂₅₇₄ lies on the line at infinity.

X_2575 — ISOGONAL CONJUGATE OF X_1114

X₂₅₇₅ISOGONAL CONJUGATE OF X₁₁₁₄

As the isogonal conjugate of a point on the circumcircle, X₂₅₇₅ lies on the line at infinity.

X_2576 — X_2-ISOCONJUGATE OF X_2574

X₂₅₇₆-ISOCONJUGATE OF X₂₅₇₄

X₂₅₇₆ = isogonal conjugate of X₂₅₈₂

X_2577 — X_2-ISOCONJUGATE OF X_2575

X₂₅₇₇-ISOCONJUGATE OF X₂₅₇₅

X₂₅₇₇ = isogonal conjugate of X₂₅₈₃

X_2578 — X_2-ISOCONJUGATE OF X_1113

X₂₅₇₈-ISOCONJUGATE OF X₁₁₁₃

X₂₅₇₈ = isogonal conjugate of X₂₅₈₀

X_2579 — X_2-ISOCONJUGATE OF X_1114

X₂₅₇₉-ISOCONJUGATE OF X₁₁₁₄

X₂₅₇₉ = isogonal conjugate of X₂₅₈₁

X_2580 — X_6-ISOCONJUGATE OF X_2574

X₂₅₈₀-ISOCONJUGATE OF X₂₅₇₄

X₂₅₈₀ = isogonal conjugate of X₂₅₇₈

X_2581 — X_6-ISOCONJUGATE OF X_2575

X₂₅₈₁-ISOCONJUGATE OF X₂₅₇₅

X₂₅₈₁ = isogonal conjugate of X₂₅₇₉

X_2582 — X_6-ISOCONJUGATE OF X_1113

X₂₅₈₂-ISOCONJUGATE OF X₁₁₁₃

X₂₅₈₂ = isogonal conjugate of X₂₅₇₆

X_2583 — X_6-ISOCONJUGATE OF X_1114

X₂₅₈₃-ISOCONJUGATE OF X₁₁₁₄

X₂₅₈₃ = isogonal conjugate of X₂₅₇₇

X_2584 — X_4-ISOCONJUGATE OF X_1113

X₂₅₈₄-ISOCONJUGATE OF X₁₁₁₃

X₂₅₈₄ = isogonal conjugate of X₂₅₈₆

X_2585 — X_4-ISOCONJUGATE OF X_1114

X₂₅₈₅-ISOCONJUGATE OF X₁₁₁₄

X₂₅₈₅ = isogonal conjugate of X₂₅₈₇

X_2586 — X_3-ISOCONJUGATE OF X_2574

X₂₅₈₆-ISOCONJUGATE OF X₂₅₇₄

X₂₅₈₆ = isogonal conjugate of X₂₅₈₄

X_2587 — X_3-ISOCONJUGATE OF X_2575

X₂₅₈₇-ISOCONJUGATE OF X₂₅₇₅

X₂₅₈₇ = isogonal conjugate of X₂₅₈₅

X_2588 — X_3-ISOCONJUGATE OF X_1113

X₂₅₈₈-ISOCONJUGATE OF X₁₁₁₃

X₂₅₈₈ = isogonal conjugate of X₁₈₂₂

X_2589 — X_3-ISOCONJUGATE OF X_1114

X₂₅₈₉-ISOCONJUGATE OF X₁₁₁₄

X₂₅₈₉ = isogonal conjugate of X₁₈₂₃

X_2590 — X_2-ISOCONJUGATE OF X_1381

X₂₅₉₀-ISOCONJUGATE OF X₁₃₈₁

X₂₅₉₀ = X₂₁₇₀-cross conjugate of X₂₅₉₁

X_2591 — X_2-ISOCONJUGATE OF X_1382

X₂₅₉₁-ISOCONJUGATE OF X₁₃₈₂

X₂₅₉₁ = crosspoint of X₅₇ and X₁₃₈₂

X_2592 — X_6-ISOCONJUGATE OF X_1822

X₂₅₉₂-ISOCONJUGATE OF X₁₈₂₂

X₂₅₉₂ = polar conjugate of X₁₁₁₃

X_2593 — X_6-ISOCONJUGATE OF X_1823

X₂₅₉₃-ISOCONJUGATE OF X₁₈₂₃

X₂₅₉₃ = trilinear pole of line X₅₂₃X₁₃₁₂