Assignment # 8

Fall 2002 Semester

GSP - Alttitudes and Orthrocenters

**Altitudes
and Orthocenters
**

**Problem 8.10:
Examine the triangle formed by the points where the extended altitudes
meet the circumcircle.
**

**
**

**How is this triangle
(triangle CDF) related to the Orthic triangle? (triangle ABE):
**

**It appears to be a
similar triangle, with each side twice the length of it’s corresponding side
of the Orthic triangle.
**

**i.e. AB/CD = BE/DF =
EA/FC = 1/2
**

**Proof:
**

**1.
Construct the nine point circle for triangle MJL.
Note that the vertices for triangle ABE fall on the nine point circle (by
definition).
**

**
**

**3.
This means that the line segments between the orthocenter and the
circumcircle are bisected by the nine point circle.
**

**
**

**4.
Length of AK/CK = BK/DK = EK/FK = 1/2
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**5.
Therefore triangle BKE is similar to triangle DKF, with
**

**
**

**6.
Therefore BE/DF = 1/2
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**7.
Similarly, AB/CD = AE/CF = 1/2
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**
**

**8.
Since all the segments between triangle ABE and triangle CDF are all
ratio ½ to each other, then the two triangles are similar.
Since all the lengths are related by a factor of 1 to 2, the triangle
areas are related by the square of the linear relationship, i.e. the CDF is 4
times the are of triangle ABE.
**

**Will the
relationship hold, if the the original triangle is obtuse?
**

**The relationship
doesn’t hold as we make the original triangle obtuse.
We lose the relationship as the perpendicular falls exterior to the
triangle.
**