The orthocenter of a triangle is the point of intersection of any two of three altitudes of a triangle (the third altitude must intersect at the same spot). The orthocenter can also be considered as a point of concurrency for the supporting lines of the altitudes of the triangle.
Let's build the orthocenter of the ABC triangle in the next app. [justify][/justify][list][*]The line segment needs to intersect point [b]C [/b]and form a right angle (90 degrees) with the "suporting line" of the side AB. Definition of "supporting line: The supporting line of a certain segment is the line[/*][*]which contains that segment" The first thing to do is to draw the "supporting line". Enable the tool LINE (Window 3) and click on points [b]A [/b]and [b]B.[/b] A line [b]d [/b]will appear.[br][/*][*]Enable the tool PERPENDICULAR LINE (Window 4), click on vertex [b]C [/b]and then click on the supporting line [b]d [/b]or on side c¹.[/*][/list][size=50][size=85]¹ In order to determine the concurrency of the orthocenter, the only important thing is the supporting line. Determining the foot of the altitude over the supporting line of the opposite side to the vertex is not necessary. However, the altitude, foot of the altitude and the supporting line of the altitude must be shown. For this reason, the supporting line of a side must always be drawn before the perpendicular line.[/size][/size][list][*]Select the tool INTERSECT (Window 2). Click on the lines [b]d [/b]and [b]e[/b]. The point [b]D [/b]will be appear.[/*][/list][b]Note: [/b]The segment CD is called the altitude of the triangle, because it connects a vertex in a perpendicular way (forming a 90 degree angle) to the supporting line of the opposite side (of the vertex). Line [b]e [/b]is the supporting line to the altitude relative to side [b]AB[/b].[br][br][img]https://www.geogebra.org/resource/ew7hyntp/vvHQ3lSy85aK6m6A/material-ew7hyntp.png[/img][br][list][*]The altitude relative to side [b]BC[/b] must be drawn now. First, let's draw the supporting line relative to side [b]BC[/b]. Enable the tool LINE (Window 3) and click on vertices [b]C [/b]and [b]B.[/b] A line “[b]f” [/b]will appear. [/*][*]Enable the tool PERPENDICULAR LINE (Window 4), click on vertex [b]A [/b]and then on the line[b] “f”[/b] or on side [b]“a”[/b]. [/*][*]Enable the tool INTERSECT (Window 2), click on line [b]f [/b]and then on line [b]g[/b]. A vertex [b]E [/b]will appear. This vertex is the foot of the altitude relative to side [b]BC[/b]. Line [b]g [/b]is the supporting line of that altitude. [/*][*]Now there are two supporting lines to the altitudes, correct? One relative to side [b]AB [/b]and the other relative to side [b]BC[/b]. Shall we mark where these lines intersect? With the tool INTERSECT TWO OBJECTS (Window 2) still enabled, click on line [b]e [/b](supporting line to the altitude relative to [br]side [b]AB[/b]) and on line "[b]g"[/b]; (supporting line to the altitude relative to side [b]BC[/b]). A new point will appear (point [b]F[/b]). This point is the orthocenter of the triangle. [/*][*]Enable the tool MOVE GRAPHICS VIEW (Window 11) to adjust the position of the objects in[br]the Viewing Window and use the [i]mouse [/i]to adjust the [i]zoom[/i]. If all the procedures were followed correctly, you will have a figure similar to the one below. [/*][/list][img]https://www.geogebra.org/resource/kkbztjje/syZ4A4baS2jE0DYP/material-kkbztjje.png[/img][list][*]The altitude relative to side [b]BC[/b] must be drawn now. First, let's draw the supporting line relative to side [b]AC[/b]. Enable the tool LINE (Window 3) and click on vertices [b]A [/b]and [b]C.[/b] A line “[b]h” [/b]will appear.[/*][*]Enable the tool PERPENDICULAR LINE (Window 4), click on vertex [b]B [/b]and then on the line[b] “h”[/b] or on side [b]“b”[/b]. A line [b]i [/b]will appear. Does this line go through the point [b]F [/b](orthocenter)? From this moment point [b]F [/b]will be [i]called [/i][b]orthocenter. [/b] In order to do this, right click the [i]mouse [/i]on[br]vertex [b]F [/b]and select the option RENAME. In the new window that appears, [b]type [/b]orthocenter and click OK.[/*][/list]
When will the triangle have an internal orthocenter? When will the triangle have an external orthocenter? When will the orthocenter coincide with one of the vertices?[br][br][br]
The supporting lines of the altitudes of a triangle intersect at the same point. [br][br]
If the orthocenter would lie outside the triangle, would the theorem proof be the same?
Move the vertices of the previous triangle and observe the angle formed by the altitudes. When will this angle be acute? When will this angle be obtuse?