De Longchamps point
- The title of this article has been capitalized due to technical restrictions. The correct title should be de Longchamps point.
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The de Longchamps point () is the the orthocenter () reflected through the circumcenter (). |
The de Longchamps point of a triangle is the reflection of the triangle's orthocenter through its circumcenter.
The point is collinear with the orthocenter and circumcenter.
De Longchamps point
Definition 1
The De Longchamps’ point of a triangle is the radical center of the power circles of the triangle. Prove that De Longchamps point lies on Euler line.
We call A-power circle of a the circle centered at the midpoint point with radius The other two circles are defined symmetrically.
Proof
Let and be orthocenter, circumcenter, and De Longchamps point, respectively.
Denote power circle by power circle by WLOG,
Denote the projection of point on
We will prove that radical axes of power and power cicles is symmetric to altitude with respect Further, we will conclude that the point of intersection of the radical axes, symmetrical to the heights with respect to O, is symmetrical to the point of intersection of the heights with respect to
Point is the crosspoint of the center line of the power and power circles and there radical axis. We use claim and get:
and are the medians, so
We use Claim some times and get: radical axes of power and power cicles is symmetric to altitude with respect
Similarly radical axes of power and power cicles is symmetric to altitude radical axes of power and power cicles is symmetric to altitude with respect Therefore the point of intersection of the radical axes, symmetrical to the heights with respect to is symmetrical to the point of intersection of the heights with respect to lies on Euler line of
Claim (Distance between projections)
Definition 2
We call circle of a the circle centered at with radius The other two circles are defined symmetrically. The De Longchamps point of a triangle is the radical center of circle, circle, and circle of the triangle (Casey – 1886). Prove that De Longchamps point under this definition is the same as point under Definition 1.
Proof
Let and be orthocenter, centroid, and De Longchamps point, respectively. Let cross at points and The other points are defined symmetrically. Similarly is diameter
Therefore is anticomplementary triangle of is orthic triangle of So is orthocenter of
as desired.
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See Also
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