Construction of a Triangle from Circumcenter, Orthocenter and Incenter

Jack D'Aurizio
30 September 2008

Looking at the The many ways to construct a triangle page I was asking myself how to find the vertices of ABC, with straightedge and compass, knowing the positions of O, H, I.

(Incidentally, I underline that (A, a, R) does not fix a triangle, since the sine law holds.)

There are some remarkable facts. By calling N the midpoint of OH we have that

  1. IN = R/2 - r due to Feuerbach's theorem.
  2. OI² = 2R (R/2 - r) due to Euler's formula.

so we know (R, r) and can easily draw the circumcircle, the incircle, the 9-points circle, the polar circle, the ortic axis and so on.

Moreover, due to Poncelet's lemma, if we take a point U on the circumcircle, draw the tangents to the incircle and intersect them with the circumcircle, we determine a chord VW that is also tangent to the incircle. (Incidentally, I've noted that one of the proofs of the Butterfly theorem on this site proves Poncelet's lemma too.)

For all these UVW triangles, I, O, R and r are fixed, so, for example, when P travels on the circumcircle, the nine-point-center N(UVW) travels on a circumference with center I and radius (R/2 - r). However, the map that brings U into N(UVW) is not a "simple" map between circumferences: it is 3-to-1, and its inversion is a cubic problem, generally not solvable with straightedge and compass.

Otherwise, we can build G (centroid), Na (Nagel point) and F (tangency point of the incircle and the 9-point-circle) and try to intersect the Feuerbach hyperbola (we know its center F and three points on it: I, H, Na) with the circumcircle, but this is a fourth (or third?!) degree problem.

This considerations raise the question of the possibility, or, more probably, the impossibility of the construction.

Using trilinears, one may say that it is hard to determine unsymmetric things like the coordinates of the vertices A(1, 0, 0), B(0, 1, 0), C(0, 0, 1) dealing with proper (symmetric) centers of the triangle (there are also a good number of reformulations in terms of trigonometric functions of the angles in A, B, C, or symmetric functions of the lengths of the sides).

So things become even harder, and I ask you:

  1. Does the (I, O, H) problem admit a simpler reverse construction (lower degree) than mine?
  2. WHEN, given three proper centers of ABC, is it possible to find (A, B, C) with straightedge and ruler?

      The Orthocenter

      1. Count the Orthocenters
      2. Distance between the Orthocenter and Circumcenter
      3. Circles through the Orthocenter
      4. Reflections of the Orthocenter
      5. CTK Wiki Math - Geometry - Reflections of the Orthocenter
      6. Orthocenter and Three Equal Circles
      7. A Proof of the Pythagorean theorem with Orthocenter and Right Isosceles Triangles
      8. Reflections of a Line Through the Orthocenter
      9. Equal Circles, Medial Triangle and Orthocenter
      10. All About Altitudes
      11. Orthocenters of Two Triangles Sharing Circumcenter and Base
      12. Construction of a Triangle from Circumcenter, Orthocenter and Incenter
      13. Reflections of the Orthocenter II
      14. Circles On Cevians

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