The Orthocenter
In this project, you will prove that all of the altitudes of a triangle intersect at a single
point: the orthocenter.
Chords in a Circle
As part of the proof for the orthocenter, we will need the following fact about two chords
that intersect in a circle:
Theorem: If chords AB and CD of a circle intersect at point X, then
mAXC is equal to the average of the measures of arcs AC and BD.
Follow this outline to prove this theorem:
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Draw chord AD .
Note that mDAB is one-half the measure of arc BD. (Use Theorem 1 from
Section 4.5.)
Note that mCDA is one-half the measure of arc AC for the same reason.
Use the fact that the sum of the angles of AXD is 180 to complete the proof.
Drawing Altitudes
An altitude of a triangle is a line that passes through one of the vertices of the triangle
and is perpendicular to the opposite side. If one of the angles of the triangle is obtuse, the
altitude may not actually intersect the opposite side.
In the examples below, note that the three altitudes intersect at a single point, even when
that point is outside of the triangle. Using Sketchpad, draw some examples of your own.
Acute-Angled Triangles
For triangles in which all of the angles are acute, the problem is a lot easier, so we’ll
restrict our attention to those kinds of triangles.
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Using Sketchpad, draw a triangle ABC and measure the angles to make sure
that they are all acute (measuring less than 90 degrees).
There is a strong relationship between the altitudes of a triangle and the circumcircle of
the triangle, which is the circle passing through the three vertices of the triangle. Recall
from Problem 20 in Section 3.3 that the center of the circumcircle is the point of
intersection of the perpendicular bisectors of the sides of the triangle.
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Using Sketchpad, draw the circumcircle of ABC using
perpendicular bisectors. Once the circle is constructed, you
should hide everything except for the original triangle and the
circle.
Now drop an altitude from point A to side BC . Let D be the intersection of this altitude
with BC , and let X be the intersection of this altitude with the circumcircle. Note that we
know the altitude will actually intersect BC since B and C are acute.
Notice from the picture that AD > DX. Show that this always
occurs as long as A is acute.
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Now construct the point H on AX such that H-D-X and
HD = DX.
We will now work to prove that BH and CH are altitudes of the
triangle, proving that all three altitudes pass through H. Note that if we prove that BH is
an altitude, the same argument will work for CH .
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Draw BH , which intersects AC at E and the
circumcircle at Y.
Since BH clearly passes through B, all that remains is to prove
that AEB is a right angle. Follow this outline to complete
the proof:
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Prove that HBD  XBD and conclude that minor arc
XC is equal in measure to minor arc YC. Use Theorem 1 from Section 4.5.
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From the Theorem about two chords in a circle, we know that mADB is the
average of the measures of arcs AB and XC.
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From the Theorem about two chords in a circle, we know that mAEB is the
average of the measures of arcs AB and YC.
Further Exploration
We have not considered the case where one of the angles of ABC is obtuse. Explore
this case, using ideas from the previous argument. Use the picture and hints below to
guide you.
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Start with ABC and draw the circumcircle
and altitude from A, as before. Note that this
time, AD < DX.
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Define point H so that D-A-H and DH = DX.
As before, point H will be the orthocenter of
this circle. We must show that BE  AC .
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First prove that BDH  BDX .
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Once again using Theorem 1 from Section 4.5,
prove that arc PAC is equal in measure to arc
CX.
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Using the Theorem about two chords in a circle, we know that mADB is the
average of the measures of arcs BPA and CX.
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There is a similar theorem to the one about two chords in a circle that shows that
mBEC is one-half the difference between the measures of arcs BXC and AP.
Find a proof of this fact.
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Use the facts that you have collected (together with the fact that the arc measure
of an entire circle is 360), finish the proof of this theorem.