All about Symmedians

This page is intended to catalog the facts concerning symmedians in a triangle, otherwise scattered throughout this site. In a reference triangle ABC, the symmedian AS_a is a cevian through vertex A (so that S_a is a point on the side line BC) isogonally conjugate to the median AM_a, M_a being the midpoint of BC. The other two symmedians BS_b and CS_c are defined similarly. The three symmedians AS_a, BS_b and CS_c concur in a point commonly denoted K and variably known as the symmedian, Lemoine or Grebe's point.

1. S_a divides BC in the ratio

   (1)	BSa / SaC = c² / b².

   
   

2. A symmedian through a vertex is the locus of the midpoints of the antiparallels to the side opposite the vertex. In particular, a symmedian splits in half a side of the orthic triangle.

3. For a point P on AS_a, the distances x and y to the sides AB and AC are in proportion to the side lengths:

   (2)	x/y = AB/AC (= c/b in standard notations).

   To see this, assume S_aD and S_aE are perpendicular to AB and AC, respectively. Then

   Area( ΔASaB) / Area( ΔASaC) = AB·SaD / AC·SaE   = BSa / SaC   = AB² / AC²,

   by (1). (2) follows directly.

   The converse is also true: any point P that satisfies the above proportion lies on AS_a.

4. The distance from the Lemoine point to the side lines of a triangle is proportional to the corresponding side lengths. This property uniquely determines the Lemoine point.

5. The Lemoine point is thus given by a : b : c in trilinear coordinates and by a² : b² : c² in the barycentric coordinates.

6. In a right triangle, the Lemoine point coincides with the midpoint of the altitude to the hypotenuse. In particular, the altitude to the hypotenuse is also the symmedian through the right angle.

7. In ΔABC the antiparallels to sides AB and AC that meet on the symmedian from C have equal lengths. By transitivity, the three antiparallels through the symmedian point all have equal lengths.

8. A symmedian through one of the vertices of a triangle passes through the point of intersection of the tangents to the circumcircle at the other two vertices. (For this reason, the symmedian always lies between the angle bisector and the altitude from the same vertex.) If A'B'C' is the tangential triangle with A' opposite A, etc. then AA', BB', CC' are concurrent since they serve as the symmedians of ΔABC.

9. The Lemoine point of a triangle also serves as the Gergonne point of its tangential triangle.

10. The Lemoine point is the center of homothety of the triangle in Grebe's construction and the reference ΔABC.

11. The centroid G of the reference ΔABC coincides with the Lemoine point of its antipedal triangle.

12. As a corollary, the Lemoine point of a triangle serves as the centroid of its pedal triangle.

13. The Lemoine point of the Gergonne triangle serves as the Gergonne point of the base triangle.

    This is because the incircle of the base triangle is the circumcircle of the associated Gergonne triangle, so that the base triangle is the tangential triangle for the associated Gergonne triangle.

14. Among all points internal to ΔABC, K minimizes the sum of the squares of the distances to the sides of the triangle.

    This follows immediately from the Lagrange identity:

    (a² + b² + c²)(x² + y² + z²) = (ax + by + cz)² + (bz - cy)² + (cx - az)² + (ay - bx)².

    Assuming, as before, that x, y, z are the distances from a point P to the sides of ΔABC and a, b, c, its side lengths, the quantity ax + by + cz represents twice the area of ΔABC and hence is constant. So is the quantity a² + b² + c². The quantity x² + y² + z², therefore, attains its minimum wherever (bz - cy)² + (cx - az)² + (ay - bx)² does. But the latter is non-negative and becomes 0 for x : y : z = a : b : c, i.e., exactly at the Lemoine point. (The requirement for P to be internal to the triangle is easily removed by introducing signed segments and areas so that some of the terms in ax + by + cz may be negative.)

15. The above has the following consequence: Of all triangles inscribed in a given triangle, that for which the sum of the squares of the sides is a minimum is the pedal triangle of the symmedian point [Johnson, p. 217].

References

1. F. G.-M., Exercices de Géométrie, Éditions Jacques Gabay, sixiéme édition, 1991
2. R. Honsberger, Episodes in Nineteenth and Twentieth Century Euclidean Geometry, MAA, 1995.
3. R. A. Johnson, Advanced Euclidean Geometry (Modern Geometry), Dover, 1960
4. V. V. Prasolov, Essays On Numbers And Figures, AMS, 2000

Symmedian

1. All about Symmedians
2. Symmedian and Antiparallel
3. Symmedian and 2 Antiparallels
4. Symmedian in a Right Triangle
5. Nobbs' Points and Gergonne Line
6. Three Tangents Theorem
7. A Tangent in Concurrency
8. Symmedian and the Tangents
9. Ceva's Theorem
10. Bride's Chair
11. Star of David
12. Concyclic Circumcenters: A Dynamic View
13. Concyclic Circumcenters: A Sequel

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