Download Orthocenter for an obtuse triangle

Use Maple to show by discovery that the three altitudes of an obtuse triangle are
concurrent. The following work was prepared in part by a graduate student in 92.527.
The three altitudes of the triangle T are drawn and the orthocenter (intersection point of
the altitudees) is labeled H. The vertices are labeled A1, A2, A3. These items are
constructed with the following comand line:
> with (geometry):
>
triangle(T,[point(A1,8,5),point(A2,0,0),point(A3,18,0)]):al
titude(hA1,A1,T):altitude(hA2,A2,T):altitude(hA3,A3,T):orth
ocenter(H,T):draw([T,hA1,hA2,hA3,H],printtext=true);
The sides of the triangle and the three altitudes are defined using the vertices and
orthocenter as the lines l1, l2, l3, l4, l5 and l6 as follows:
>
line(l1,[A1,A2]):line(l2,[A2,A3]):line(l3,[A1,A3]):line(l4,
[A1,H]):line(l5,[A2,H]):line(l6,[A3,H]);
l6
The figure is redrawn to include point B, the intersection of the altitude from vertex A1 to
the base of the triangle.
>
intersection(B,l4,l2):draw([T,hA1,hA2,hA3,H,B],printtext=tr
ue);
The angles between the lines are evaluated. Note that the obtuse angle at vertex A1 is
displayed as the smaller of two supplementary angles.
> evalf(convert(FindAngle(l1,l2),degrees));
32.00538321 degrees
> evalf(convert(FindAngle(l2,l3),degrees));
26.56505117 degrees
> evalf(convert(FindAngle(l1,l3),degrees));
58.57043436 degrees
Since the sum of the interior angles of a triangle sum to 180 degrees, the obtuse angle at
vertex A1 is actually
> evalf(convert(Pi,degrees)convert(FindAngle(l1,l3),degrees));
121.4295656 degrees
Let us check if lines l1, l2, l3 are concurrent, which they shouldn't be.
> AreConcurrent(l1,l2,l3);
false
What about lines l4, l5 and l6. Are they concurrent?
> AreConcurrent(l4,l5,l6);
true
The coordinates of point H are:
> coordinates(H);
[ 8, 16 ]
An interesting observation in the previous drawing is the angle at the vertex H of the
triangle formed by vertices H-A2-A3. It is the supplement of the angle at vertex A1.
> evalf(convert(FindAngle(l5,l6),degrees));
58.57043436 degrees
How come this occuurs? Is it by chance or can we show why? Construct the quadrilateral
A1-C-H-D such that the line extending through A1 and A3 intersects the line through A2
and H at point C. Likewise point D is the intersection of the extension of the line through
points A2 and A1 with the line through A3 and H. To see what it looks like would
involve more work in Maple, but analytically the angles formed at points C and D are
right angles since they occur at the intersections of triangle sides and altitudes. We then
have a quadrilateral with two right angles as opposite angles. Therefore the other two
angles must be supplementary in order for the sum of the four interior angles of the
quadrilateral to equal 360 degrees.
Summary: Geometry work with Maple assumes we are familiar with terminology.
Editing and what-if scenarios at times cause redefinition of Maple commands.