Matpic
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Series

Triangle centres — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Nine points of a triangle, on one circle. A triangle with the midpoints of its sides, the feet of its three altitudes and the midpoints from each corner to the orthocentre marked; all nine lie on a single circle of half the circumradius.

    Nine points on one circle

    Three midpoints, three feet of altitudes and three more midpoints. Nine points defined in three unrelated ways, on an arbitrary triangle, and all nine sit on one circle — checked here on two hundred and forty triangles as well as on the drawn one.

    part 1 · geometry
  2. The nine-point circle, touching four others. A triangle with its nine-point circle, its inscribed circle and its three escribed circles, each of the four tangent to the first — with the distances between centres compared against the radii.

    One circle touching four

    The nine-point circle touches the inscribed circle and each of the three escribed ones. Nothing in its construction mentions them, the two families of centres are built from different kinds of number, and the tangency is four exact equalities between distances and radii.

    part 2 · geometry
  3. The classical centres as three weights each. A table of triangle centres with the weights on the three corners that produce each, and the determinants that decide which triples of them are collinear.

    A centre is three weights

    Write each classical centre as a weighted average of the corners and a coincidence becomes a determinant. The Euler line is then one number rather than a construction, the whole catalogue becomes mechanical, and the reason one centre is missing from it is visible in the weights.

    part 3 · geometry
  4. Three lines through a point, reflected in the bisectors, meet again. A triangle with a point P and its cevians, their reflections in the angle bisectors, and the point P* where the reflections meet — P's isogonal conjugate, at (373.8, 290.0).

    Every point has a partner across the bisectors

    Draw the three lines from the corners of a triangle through any point, reflect each in the bisector of its own angle, and the three reflections meet again. The pairing this makes swaps the centroid with the symmedian point and the orthocentre with the circumcentre, bends every straight line into a conic through the corners, and sends the circumcircle to infinity.

    part 4 · geometry
  5. Five triangles between the same two circles, every one closing. An outer circle of radius 1 and an inner of radius 0.38 at Euler's distance 0.4899; five triangles inscribed in the first and circumscribed about the second, started from different points.

    A triangle that fits once fits everywhere

    Put one circle inside another and try to fit a triangle between them, its corners on the outer circle and its sides touching the inner. Usually no triangle fits. But if one does, then one fits starting from every point of the outer circle — and whether it does is decided by a single equation in the two radii and the distance between the centres.

    part 5 · geometry

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