The Orthocenter
Michael Keyton
Illinois Mathematics and Science Academy
1500 W. Sullivan Rd.
Aurora, IL 60506
[email protected]
Orthocenter
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Keyton
The Orthocenter
Michael Keyton
Illinois Mathematics and Science Academy
The orthocenter is the intersection of the altitudes of a triangle. Of the four classic
centers discovered by the ancient Greeks, the orthocenter has more results known about it
than any of the others. Since the orthocenter is intimately related to the orthic triangle and
to orthocentric sets, theorems about them will also be included. There is no claim that this
is anywhere near a complete list of the theorems about the orthocenter, but is offered as a
beginning. The theorems that appear here are given without proof, but have been
organized into categories by the author.
Only the incenter and the circumcenter appear in Euclid. Proclus proved the
orthocenter exists; however, it had been mentioned indirectly in the writings of
Archimedes, explaining why it has also been called the Archimedean point. William
Henry Besant (1828-1917) first used the word orthocenter in Conic sections, treated
geometrically (1869). In Mechanica, Heron proved the centroid exists, which
Archimedes also knew existed.
The orthocenter will be designated by H, the circumcenter by O, the incenter by I, and
the centroid by G.
I.a Distances
(1) The products of the segments of an altitude determined by the orthocenter are equal.
(2) The product of the projections of two sides of a triangle onto the third side is equal to
the product of the altitude to that side and the distance of the orthocenter from the side.
(This generalizes a major theorem of a right triangle and the altitude to the hypotenuse.)
(3) The sum of the squares of the sides of the triangle and of the distances from the
orthocenter to the vertices of a triangle is twelve times the square of the circumradius.
(4) The sum of the radii of the six circles formed by the altitudes and the sides of a
triangle and semiperimeter is equal to the sum of the altitudes of the triangle.
(5) The sum of the distances from the orthocenter to the feet of the altitudes is four times
the sum of the circumradius and the inradius.
(6) The sum of the squares of the distances from the orthocenter to the feet of the
altitudes is four times the product of the circumradius and the difference of the
circumradius and the inradius.
I.b Formulae Relating the Four Centers
(7) HI2 + 2OI2 = 3(IG2 + 2OG2)
(8) 3(HG2 + 2IG2) = 2IH2 + 4R(R-2r), where R = circumradius and r = inradius
II. Complete Quadrilateral
(1) The orthocenters of the four triangles of a complete quadrilateral are collinear.
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III. The Orthic Triangle, Orthic Axis, or Tangential Triangle
(0) q.v., VI.2, VI.6
(1) The orthic triangle is the triangle with minimum perimeter that can be inscribed in an
acute angled triangle.
(2) The pedal triangle of the orthocenter is its cevian triangle, the orthic triangle.
(3) The sides of the orthic triangle cut off triangles similar to the triangle.
(4) The orthic triangle and the tangential triangle (triangle formed by the tangents to the
circumcircle at the vertices) are homothetic (at the Gob point, located on the Euler line,
k = R/p, where R is the circumradius and p is the inradius of the orthic triangle).
(5) The product of the segments of a side of a triangle determined by the orthic triangle is
equal to the product of the sides of the orthic triangle intersecting on the side.
(6) The product of the six segments of the sides of a triangle determined by the orthic
triangle is equal to the square of the products of the sides of the orthic triangle.
(7) If P and Q are the projections of B and C of ΔABC onto sides A'C' and A'B' of orthic
ΔA'B'C', then PC' = QB'.
(8) If P and Q are projections of A', the foot of altitude AA', onto sides AB and AC of
ΔABC, then P, Q, B, and C are concyclic.
(9) The perimeter of the orthic triangle of an acute angled triangle is less than twice the
smallest altitude.
(10) The ratio of a side of the triangle to the corresponding side of the orthic triangle is
equal to the ratio of the circumradius and the distance of the side from the circumcenter.
(11) Given a side and the opposite angle, the corresponding side of the orthic triangle is
fixed.
(12) The perimeter of the orthic triangle of an acute angled triangle is equal to twice the
area of the triangle divided by the circumradius.
(13) In an acute angled triangle, the sum of the distances of the vertices from the sides of
the orthic triangle is equal to twice the circumradius increased by the distance from the
orthocenter to a side of the orthic triangle.
(14) Perpendiculars from the midpoints of the orthic triangle to the opposite sides of the
triangle are concurrent.
(15) Perpendiculars from the midpoints of the triangle to the corresponding sides of the
orthic triangle are concurrent at the nine-point center.
(16) The sides of the orthic triangle intersect the sides of the triangle again in three
collinear points, the line is called the orthic axis. (follows from Desargues' Theorem)
(17) If the sides of the cevian triangle of a point are antiparallel with respect to the sides
of the triangle, then the point is the orthocenter.
(18) The foot of a perpendicular from the orthocenter to a line joining a vertex to the
intersection of the opposite side of the triangle and the corresponding side of the orthic
triangle is on the circumcircle.
(19) The orthic axis is the trilinear pole of the orthocenter with respect to the triangle and
the orthic triangle.
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IV. Orthocentric Sets
(1) Given the vertices of a triangle and the orthocenter, the orthocenter of any triangle
formed by any three of these points is the fourth point. Four points satisfying this
condition is called an orthocentric set.
(2) The four triangles of an orthocentric set have the same orthic triangle.
(3) The four triangles of an orthocentric set have the same nine-point circle.
(4) The circumcircles of the four triangles in an orthocentric set are congruent.
(5) The circumcenters of an orthocentric set form an orthocentric set. They are the
symmetrics of the points with respect to the nine-point center.
(6) The centroids of an orthocentric set form an orthocentric set.
(7) The tritangent centers (the incenter and the excenters) is an orthocentric set.
(8) The nine-point circle of the triangles of an orthocentric set is the nine-point circle for
the orthocentric set of circumcenters.
(9) The nine-point circle of the triangles of an orthocentric set is the nine-point circle for
the orthocentric set of centroids.
(10-11) From (8-9), the orthocentric set of centroids and the orthocentric set of
circumcenters are homothetic with the orthocentric set (center the Nine-point center, k = 1/3, -1)
(12) The Euler lines of the four triangles of an orthocentric set are concurrent.
(13) The circumcenters of the three triangles formed using two vertices and the
orthocenter is a triangle congruent (and homothetic) to the original triangle.
(14) The orthocenter of the triangle is the circumcenter of the triangle in (13).
(15) The circumcenter of the triangle is the orthocenter of the triangle in (13).
(16) The centroids of the three triangles formed using two vertices and the orthocenter is
a triangle homothetic to the original triangle.
(17) The orthocenter is the centroid of the triangle in (16).
(18) Using directed distances, the sum of the distances from the vertices of an
orthocentric set to a line through the nine-point center is 0.
(19) When these theorems are applied to the orthocentric set formed by the excenters, a
large set of theorems result.
(20) The incenter and the circumcenters of the triangles formed using two vertices of a
triangle and the incenter form an orthocentric set.
(21) The circumcenter and the circumcenters of the triangles formed by two vertices of a
triangle and the circumcenter form an orthocentric set.
V. The Centroid, Medial Triangle, or Anticomplementary Triangle (Exmedial)
(0) q.v., IV.6, 9, 10, 16, 17; VI.7, VIII.18, XI.2
(1) The line joining the centroid to a point on the circumcircle bisects the segment joining
the orthocenter and the antipode of the point.
(2) Three congruent circles with centers the vertices of a triangle intersect the respective
sides of the medial triangle in six points equidistant from the orthocenter.
(3) The centroid and the orthocenter are the centers of similitude of the circumcircle and
the nine-point circle of a triangle.
(4) The circle with diameter the centroid and the orthocenter is coaxal with the
circumcircle and the nine-point circle.
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VI. The Incenter, Excenters, Incircle, or Spieker Point
(0) q.v., IV.7, 19, 20; VII.6
(1) The orthocenter is the incenter of its pedal triangle in an acute triangle. (in an obtuse
triangle, it is an excenter) H(Δ)=I(Pt(H,Δ))
(Equivalently: The incenter of the orthic triangle is the orthocenter of the triangle.)
(2) The sides of the triangle bisect the exterior angles of the orthic triangle.
(3) The incenter is the orthocenter of the triangle formed by the excenters. I(Δ) =
H(exc(Δ))
(4) The incenter is the orthocenter of the triangle formed by the midpoints of the arcs of
the circumcircle determined by the sides of a triangle.
(5) The sum of the distance from the orthocenter to a vertex and the distance from the
orthocenter to the corresponding excenter is constant (the sum of the inradius and twice
the circumradius).
(6) In an obtuse triangle, the orthocenter and the circumcenter are excenters of the orthic
triangle and the tangential triangle.
(7) The incenter of the medial triangle, the Spieker point, bisects the segment joining the
orthocenter and the circumcenter of the triangle formed by the excenters.
(8) In an acute angled triangle, the incenter of the triangle formed by the symmetrics of
the sides of the orthic triangle with respect to the corresponding sides of the triangle is the
circumcenter of the triangle.
(9) The triangle formed by the centers of the incircles of the triangles cut off by the orthic
triangle and the points of tangency of the incircle are homothetic (k=-1)
VII. The Circumcenter
(0) q.v., IV.5, 8, 11, 13, 14, 15, 21; VI.6, 7, 8; XV.5
(1) The orthocenter and circumcenter are isogonal conjugates.
(2) Equivalent to (1). Perpendiculars from the vertices to the sides of the orthic triangle
intersect at the circumcenter. The circumcenter and the orthocenter are antipedal points.
(3) The circumcenter is the orthocenter of its pedal triangle.
(4) The distance from a vertex to the orthocenter is twice the distance from the
circumcenter to the opposite side of the triangle.
(5) The symmetric of the orthocenter with respect to a vertex and the symmetric of the
vertex with respect with the midpoint of the opposite side are collinear with the
circumcenter.
(6) mA=60º iff the circumcenter, the orthocenter, the incenter, and the excenter
opposite A are concyclic. The circumcenter and the orthocenter are equidistant from the
incenter and the excenter.
(7) The circumcenter O of ΔABC is the orthocenter of ΔPQR where P, Q, and R are the
circumcenters of ΔABO, BCO, and CAO.
(8) A line through the circumcenter of ΔABC intersects BC, CA, and AB at P, Q, and R.
Circles with diameters AP, BQ, and CR are concurrent at S and T, one of which is on the
circumcircle and the other is on the nine-point circle. The orthocenter is between S and T.
(9) The reflection of the orthocenter through the midpoint of a side is the antipode of the
vertex opposite the side (reflection of the vertex through the circumcenter). (MK 1/23/02)
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VIII. The Circumcircle or the Anticenter
(0) q.v., IV.4, V.1, 3, 4; VI.4, IX.1
(1) The reflection of the orthocenter over a side of the triangle is on the circumcircle.
(2) The circles formed by two vertices of a triangle and the orthocenter is congruent to
the circumcircle.
(3) The locus of the reflection of a point on the circumcircle over a side of a triangle is a
circle congruent to the circumcircle passing through the orthocenter. Thus, the
intersection of the loci of the reflections of a point on the circumcircle over the sides of a
triangle is the orthocenter.
(4) Parallels through the intersection points of the altitudes with the circumcircle to the
respective radii to the vertices of the triangle are concurrent. (q.v., X1.2)
(5) The triangle formed by the intersection of the altitudes of a triangle and the
circumcircle is homothetic to the triangle (about the orthocenter, k=2).
(6) The vertices of the triangle are the midpoints of the arcs formed by the intersection of
the altitudes and the circumcircle.
(7) If two triangles are inscribed in a circle with a common base, the segment containing
the orthocenters of the triangles is congruent and parallel to the segment joining the noncommon vertices.
(8) The perpendicular from the orthocenter H to altitude HC of ΔABC intersects the
circumcircle of ΔHBC at P. ABPH is a parallelogram.
(9) The orthocenters of the four triangles formed by a cyclic quadrilateral form a cyclic
quadrilateral homothetic with the original quadrilateral.
(10) The lines obtained by joining each vertex of a cyclic quadrilateral to the orthocenter
of the triangle formed by the other three vertices bisect each other.
(11) The anticenter, the point of concurrency of perpendiculars from the midpoints of a
cyclic quadrilateral of a cyclic quadrilateral to the opposite sides, is the orthocenter of the
triangle having for vertices the midpoints of the diagonals and the intersection of the
diagonals.
(12) The vertices of a cyclic quadrilateral are the orthocenters of the four triangles formed
using three orthocenters of the four triangles formed using the vertices of the cyclic
quadrilateral.
(13) Parallels through the orthocenter of ΔABC to AB and AC intersect BC at D and E.
Perpendiculars to BC at D and E intersect AB and AC at D' and E' which are collinear
with the antipodal points of B and C on the circumcircle. This line is parallel to BC.
(14) The symmetrics of a point on the circumcircle with respect to the sides of the
triangle are collinear with the orthocenter.
(15) The power of the orthocenter with respect to the circumcircle of a triangle is equal to
four times its power with respect to the nine-point circle.
(16) The square of the radius of the polar circle of a triangle is equal to half the power of
the orthocenter of the triangle with respect to the circumcircle of the triangle.
(17) The sum of the powers, with respect to the circumcircle of a triangle, of the
symmetrics of the orthocenter with respect to the vertices of the triangle is equal to the
sum of the squares of the sides of the triangle.
(18) (extension discovered on 1/18/2002 by the author) VII.13 for any point on the circle
with diameter GH, center = X(381).
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IX. The Nine-Point Circle and Center
q.v., III.15, IV.8, 9, 18; V.3, 4; VII.8, VIII.15, XV.2
(1) The midpoint between a point on the circumcircle and the orthocenter is on the ninepoint circle.
(2) The locus of the midpoints in V.1 is the nine-point circle.
(3) The projections of the orthocenter of a triangle on the two bisectors of an angle of the
triangle are on the line joining the midpoint of the opposite side to the nine-point center.
(4) The symmetric of the orthopole of a line through the orthocenter is on the nine-point
circle.
X. The Euler Line
(0) q.v., III.4, IV.12
(1) The orthocenter and the circumcenter are collinear with the centroid, forming the
Euler line. The nine-point center (N) is the midpoint between the orthocenter and the
circumcenter. HG = 2OG, HN = NG
(2) The sides of the orthic triangle intersect the other sides of the triangle in three
collinear points. The line through these points is perpendicular to the Euler line.
XI. The Symmedian Point
(1) The isotomic of the orthocenter is the symmedian of the exmedial triangle.
(2) The symmedian point of the orthic triangle is collinear with the symmedian point and
orthocenter of the triangle.
(3) The four lines joining the symmedian points and orthocenters of the triangles of an
orthocentric set are concurrent.
XII. The Radical Axis and Center
(1) The radical center of the three circles with sides of a triangle as diameters is the
orthocenter.
(2) The orthocenter is on the radical axis of two circles constructed with diameters two
cevians.
(3) The radical center of the circles with diameters any three cevians of a triangle is the
orthocenter.
(4) For any three non-coaxal circles having cevians of a triangle for diameters, the
orthocenter is the radical center.
XIII. The Orthopole
(0) q.v., IX.4
(1) The orthopole of a side of a triangle with respect to the triangle is the orthocenter.
XIV. Polar Triangles
q.v., VIII.12
(1) If a triangle is polar with respect to a circle, then the center of the circle is the
orthocenter of the triangle.
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XV. The Wallace line (Simson line)
(1) The Wallace line for a point on the circumcircle bisects the segment joining the point
and the orthocenter.
(2) The locus of the midpoint in (XIV.1) is the nine-point circle.
(3) The Wallace line of P for ΔABC with orthocenter H intersects BC at L, and the
altitude AD at K, PK is parallel to LH.
(4-5) The Wallace lines of the symmetrics of the orthocenter over the sides of the triangle
form a triangle homothetic to the orthic triangle. Its circumcenter is the orthocenter of the
orthic triangle.
XVI. The Nagel Point and the Fuhrmann Circle
(1) The orthocenter and the Nagel point are antipodal points on the Fuhrmann circle.
XVII. The Brocard Circle
(1) The orthocenter lies on the Brocard circle iff the triangle is equilateral.
XVIII. Miscellaneous Problems
(1) BAMN, ACPQ, CBST are external squares on the sides of ΔABC. D, E, F are the
midpoints of MQ, PT, and SN. Lines AD, CE, and BF are concurrent at the orthocenter.
(2) Perpendiculars from the orthocenter to the lines from the vertices of a triangle to a
given point intersect the respectively opposite sides in three collinear points.
(3) Line λ intersects the sides of ΔABC at C', A', and B' respectively, perpendiculars from
A, B, C to the lines HA', HB', and HC' respectively are concurrent at a point on the
perpendicular from H to λ.
(4) Perpendiculars from the orthocenter to three concurrent cevians of a triangle intersect
the respective circles having the cevians for diameters in 6 concylic points.
(5) Perpendiculars from the orthocenter to three concurrent cevians of a triangle intersect
the respective circles having the sides of the triangle as diameters in 6 concyclic points.
(6) Given ΔABC, parallels through H to AB and AC intersect BC at D and E
respectively, perpendiculars to BC at D and E intersect AB and AC at D' and E', D'E'
intersects the circumcircle at C' and B'. C' and B' are the antipodes of C and B.
XVIV. Constructions
(1) A triangle can be constructed given (A Δ cbcg) the three altitudes.
(2) A Δ cbcg the orthocenter, a vertex, and the directions of the sides through the vertex.
(3) A Δ cbcg the orthocenter and two vertices.
(4) A Δ cbcg the feet of two altitudes and the line containing the third side.
(5) A Δ cbcg the intersection points of the altitudes and the circumcircle.
6) A Δ cbcg a vertex, the orthocenter, and the nine-point center.
(7) A Δ cbcg the orthocenter, a vertex, and the point of intersection of the opposite side
and the corresponding side of the orthic triangle.
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XX. Locus
(0) V.2, XI.2, XII.2, XV.2
(1) The locus of the orthocenter with a fixed side and the opposite vertex on a circle is a
circle, the reflection of the circumcircle over the fixed side.
(2) The locus of point in VI.8 with a side of the triangle fixed and the vertex on a circle is
an ellipse, center the circumcenter.
(3) Given perpendicular chords through a fixed point in the interior of a circle, the locus
of the orthocenters of the four triangles formed by the endpoints are the endpoints of
perpendicular chords through the fixed point in a circle with center the symmetric of the
center of the circle with respect to the fixed point.
(4) The locus of the orthocenter with a fixed vertex and fixed nine-point circle is a circle.
(5) The locus of the nine-point center with a fixed orthocenter, midpoint of a side, and
direction of the base is a line perpendicular to the base at the midpoint between the
midpoint and the projection of the orthocenter onto the base.
(6) S and T are equidistant from O, tangents SA and TB intersect at M, the locus of the
orthocenter of ΔMAB is a segment on the perpendicular bisector of ST or an arc with
center the midpoint of S and T.
REFERENCES:
Altshiller-Court, Nathan, College Geometry, Barnes & Noble, 1952.
Aref, M.N. and W. Wernick, Problems & Solutions in Euclidean Geometry, Dover, 1968.
Coxeter, H.S.M. and S.L. Greitzer, Geometry Revisited, MAA, 1967.
Davis, David R., Modern College Geometry, Addison-Wesley, 1957.
Johnson, Roger A., Advanced Euclidean Geometry, Dover, 1960.
Honsberger, Ross, Episodes in 19th and 20th Century Euclidean Geometry, MAA, 1995.
Kimberling, Clark, Triangle Centers and Central Triangles, Utilitas Math. Publ., 1998.
F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, 1991 (reprint).
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