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Elementary Geometry
 Miscellaneous Propositions
  Theorem
   Proof
   Examples
   Proof
 Sources and References

Elementary Geometry

Miscellaneous Propositions

976

Theorem

When three perpendiculars to the sides of a triangle 𝐴𝐵𝐶, intersecting them in the points 𝑎, 𝑏, 𝑐 respectively, are concurrent, the following relation is satisfied; and conversely, if the relation be satisfied, the perpendiculars are concurrent. 𝐴𝑏2−𝑏𝐶2+𝐶𝑎2−𝑎𝐵2+𝐵𝑐2−𝑐𝐴2=0

Proof

If the perpendiculars meet in 𝑂, then 𝐴𝑏2−𝑏𝐶2=𝐴𝑂2−𝑂𝐶2, ⋯ (I.47).

Examples

By the application of this theorem, the concurrence of the three perpendiculars is readily established in the following cases:
  • When the perpendiculars bisect the sides of the triangle.
  • When they pass through the vertices. (By employing I.47)
  • The three radii of the eseribed circles of a triangel at the points of contact between the vertices are concurrent. So also are the radius of the inscribed circle at the point of contact with one side, and the radii of the two escribed circles of the remaining sides at the points of contact beyond the included angle.
    In these cases employ the values of the segments given in (953).
  • The perpendiculars equidiculars from the vertices with three concurrent perpendiculars are also concurrent.
  • When the three perpendiculars from the vertices of one triangle upon the sides of the other are concurrent, then the perpendiculars from the vertices of the second triangle upon the sides of the first are also concurrent.

Proof

If 𝐴, 𝐵, 𝐶 and 𝐴′, 𝐵′, 𝐶′ are corresponding vertices of the triangle, join 𝐴𝐵′, 𝐴𝐶′, 𝐵𝐶′, 𝐵𝐴′, 𝐶𝐴′, 𝐶𝐵′, and apply the theorem in conjunction with (I.47)

Sources and References

https://archive.org/details/synopsis-of-elementary-results-in-pure-and-applied-mathematics-pdfdrive

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ID: 210900026 Last Updated: 9/26/2021 Revision: 0 Ref:

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References

  1. Hilbert, D. (translated by Townsend E.J.), 1902, The Foundations of Geometry
  2. Moore, E.H., 1902, On the projective axioms of geometry
  3. Fitzpatrick R. (translated), Heiberg J.L. (Greek Text), Euclid (Author), 2008, Euclid's Elements of Geometry
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