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 itís 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).

 2.  The circumcircle for triangle MJL is dilation of the nine point circle of MJL in the ratio of 2 to 1.

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

5.  Therefore triangle BKE is similar to triangle DKF, with

6.  Therefore BE/DF = 1/2

7.  Similarly, AB/CD = AE/CF = 1/2

8.  Since all the segments between triangle ABE and triangle CDF are all ratio Ĺ 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 doesnít hold as we make the original triangle obtuse.   We lose the relationship as the perpendicular falls exterior to the triangle.

 

 

 

 

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