The Department of Mathematics Education

 

William Daly

EMAT6680

Write Up #8

††††††††††† Summer 03

 

Altitude Relationships to the Circumcircle and Orthocenter

 

Shown below is an arbitrary acute triangle.The construction of the altitudes and circumcircle are shown.Using GSPís measurement tools, it is demonstrated that the product of the two segments of the altitude on either side of the orthocenter is the same constant value for all three altitudes of the triangle.Throughout this proof, several other similarities of the inscribed figures are discovered.

 

 

 

To prove this relationship, first consider just one of the altitudes:

 

 

 

The triangle BPC segments were included in this figure.Notice that there are several chords in this diagram subtended by two angles with vertices on the circumcircle.These angles are equal.So for chord AB ACB=APB.For chord BP BAP=BCP.For chord PC BPC=PAC.Finally for chord AC CPA=CBA.With these equal angles, it is clear that BDA@PDC and ADC @BDP.This is shown below with similar triangles shaded in the same colors.

 

 

This argument is applied to each of the perpendicular lines, leading to the following similar triangles.All of the chords are drawn in this figure.The figure replicated three times to make the similar triangles more visible.

 

 

 

There are additional similarities among these figures.For example DBA=CBF appears in both the first and third figures, and these two triangle have a right angle at points D and F respectively.So BAD=BCF.But from the similar triangles above, both of these are also equal to DCP.This argument applied to the other pairs of figures, 1,2 and 2,3, shows that the remaining triangles in white are similar to each of their reflections aboutchords AB, AC and BC, the legs of the original triangle.This also proves that the length of PD=DH, QE=EH and RF=FH.The relationships shown below follow from consider the same relationships among the similar angle between the figures above.

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Considering similar triangles from the blue, black and red congruent angles above, it is clear that:

; ;

Apply a shorthand notation to the segment lengths.Each one will be the distance from the point H.So AH becomes A, BH become B, etc.

 

Then AD=CF=BE.That is the product of the two segments of the altitude on either side of H is a constant.

 

 

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