What are the steps used to construct a triangle orthocenter?

Answer 1

1. Take the triangle and label the vertices ABC so that BC is the shortest side.
2. Take your pairs of compasses and set them to the length BC.
3. With the compass point on B draw a small arc to intersect AB and label this point D
4. With the compass point on C draw a small arc to intersect AC and label this point E
5. Now construct the angle bisectors of BCD and CBE - where these two lines meet is the orthocentre:

To construct an angle bisector:

1. Set your compasses at some length.
2. Mark a small arc on each arm of the angle; label the two points X and Y
3. With your compasses set to any length greater than half XY, on one point (say X) draw a small arc between the arms of the angle approximately in the middle - this will be the other side of the point to the vertex
4. With your compasses on the other point (say Y) draw a small arc to intersect the arc drawn in the last step.
5. Draw in the line between the two vertex and the intersection of the arcs - this is the angle bisector.

Note:

In constructing the angle bisectors, the first two steps are done in creating points D and E; the points X and Y are:

• for angle BCD: B and D as you are bisecting the angle at vertex C
• for angle CBE: C and E as you are bisecting the angle at vertex B

You can use your compasses set to the length BC for all the arcs to be drawn.

As you have used the length BC to create the points D and E, the triangles created by BCD and CBE are isosceles - in BCD the equal side are BC and BD, in CBE the equal sides are CB and CE. Thus when you draw in the angle bisectors of BCD and CBE they are perpendicular to the base of their respective triangles and thus the heights of those triangles, which is also the height of the original triangle (as point A is on an extension of BD and CE respectively).

If there is no one shortest side, either (or both) of points D and/or E will coincide with point A.