Hamilton's theorem

If in the shown figure the orthocenter of an acute triangle ABC is denoted by T , then the three Hamilton triangles TAB , TBC and TCA have a common Euler circle (a circle of nine points ).

The three Hamilton triangles in the Hamilton theorem form the so-called dragon eye .

Hamilton's theorem is used as part of the Johnson theorem (see figure).

• Three segments of lines connecting the orthocenter with the vertices of an acute triangle divide it into three Hamilton triangles with equal radii of the circumscribed circles.
• The radii of the circumscribed circles of the three Hamilton triangles are equal to the radius of the circle described around the original acute triangle. Let's call them Hamilton-Johnson circles.
• The radii of the circumscribed circles of the three Hamilton triangles have three centers J _A , J _B and J _C. These three centers form the vertices of the Johnson triangle ΔJ _A J _B J _C , which is equal to the original triangle Δ ABC and has pairs of sides parallel to it ( Johnson's Theorem , see figure).
• If through the vertices of the original triangle ABC we draw straight lines parallel to opposite sides, then we get an anti- additional triangle similar to the original triangle ABC , whose vertices P _A , P _B and P _C lie on three Hamilton-Johnson circles having equal radii (see fig.) .

Both corollaries immediately follow from Hamilton's theorem if we notice that the radius of the Euler circle is half the radius of the circle described around the same triangle.

• For an obtuse triangle, Hamilton's theorem is reformulated as follows. Let the orthocenter outside the obtuse-angled triangle, as the intersection point of its two heights, dropped from the vertices of two acute angles on the extension of its two sides, and the continuation of the third height, drawn from the vertex of the obtuse angle. Then the orthocenter and the two vertices of acute angles form an acute triangle, to which the Hamilton theorem is applicable. In particular, the obtuse triangle itself will be one of the three Hamilton triangles . The vertices of the other two Hamilton triangles are the orthocenter and the vertices of two adjacent sides, forming the obtuse angle of the obtuse triangle.
• For a right-angled triangle, the orthocenter coincides with the vertex of a right angle, and one Hamilton triangle coincides with this right-angled triangle with the correct radius (diameter) of the circumscribed circle . The remaining two Hamilton triangles degenerate into two legs at the vertex of the right angle. Through these two legs (like a triangle with two points - vertices) one can draw countless circumscribed circles with diameters not less than the length of these legs. That is, Hamilton's theorem is formally satisfied in this limiting case.

If in the shown figure the orthocenter of an acute triangle ABC is denoted by T , then for the obtuse triangle TBC the orthocenter is point A. Moving from the obtuse triangle TBC to the acute triangle ABC , Hamilton's theorem can be used again.

The theorem was proved by the outstanding Irish mathematician and nineteenth-century physicist William (William) Rowan Hamilton in 1861. Hamilton, William Rowan (1806–1865) is an Irish mathematician.

• Dm Yefremov New triangle geometry . - 1902.
• Zetel S.I. New triangle geometry. Manual for teachers. 2nd edition .. - M .: Uchpedgiz, 1962. - 153 p.
• William Rowan Hamilton http://free-math.ru/publ/istorija_matematiki/velikie_matematiki/gamilton_uiljam_rouehn/22-1-0-168

• Triangle
• Euler Circle
• Nine point circumference
• Orthocenter
• Segments and circles associated with a triangle
• Johnson's theorem