Equal Circles, Medial Triangle and Orthocenter
The applet below illustrates the following problem [Altshiller-Court, Theorem 181]:
Any three equal circles having for centers the vertices of a given triangle cut the respective sides of the medial triangle in six points equidistant from the orthocenter of the given triangle.
What if applet does not run? |
References
- N. Altshiller-Court, College Geometry, Dover, 1980
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Copyright © 1996-2018 Alexander BogomolnySolution
Any three equal circles having for centers the vertices of a given triangle cut the respective sides of the medial triangle in six points equidistant from the orthocenter of the given triangle.
Note: The problem is equivalent to saying that the six points in question are concyclic; thus it claims the existence of a circle. One can easily surmise that the circle exists even if only four of the six points are available.
Let ΔABC be given, with Ma, Mb, Mc the midpoints and the orthocenter H.
What if applet does not run? |
Assume circle centered at A meets MbMc in U and U'. Let B be the intersection of the altitude from A and MbMc. The two right triangles APU and APU' are equal. Applying the Pythagorean identity gives
PU² = PU'² = AU² - PU² = HU² - HP².
Hence,
AU² - HU² | = AP² - HP² | |
= (AP + HP)(AP - HP) | ||
= AH (DP - HP) | ||
= AH·DH, |
implying
HU² = AU² - AH·DH.
Now, by assumption, AU = BV = CW. So it follows from a property of the orthocenter that HU = HV = HW.
The Orthocenter
- Count the Orthocenters
- Distance between the Orthocenter and Circumcenter
- Circles through the Orthocenter
- Reflections of the Orthocenter
- CTK Wiki Math - Geometry - Reflections of the Orthocenter
- Orthocenter and Three Equal Circles
- A Proof of the Pythagorean theorem with Orthocenter and Right Isosceles Triangles
- Reflections of a Line Through the Orthocenter
- Equal Circles, Medial Triangle and Orthocenter
- All About Altitudes
- Orthocenters of Two Triangles Sharing Circumcenter and Base
- Construction of a Triangle from Circumcenter, Orthocenter and Incenter
- Reflections of the Orthocenter II
- Circles On Cevians
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