The orthic triangle is also homothetic to two important triangles: the triangle formed by the tangents to the circumcircle of the original triangle at the vertices (the tangential triangle), and the triangle formed by extending the altitudes to hit the circumcircle of the original triangle. This circle is better known as the nine point circle of a triangle. The circumcircle of the orthic triangle contains the midpoints of the sides of the original triangle, as well as the points halfway from the vertices to the orthocenter. The next easiest to find is the one from B B B to A C AC A C, since A C AC A C can be calculated as y = 12 5 x y=\frac n m for relatively prime positive integers m m m and n n n. The easiest altitude to find is the one from C C C to A B AB A B, since that is simply the line x = 5 x=5 x = 5. What are the coordinates of the orthocenter? This is especially useful when using coordinate geometry since the calculations are dramatically simplified by the need to find only two equations of lines (and their intersection). Finally, if the triangle is right, the orthocenter will be the vertex at the right angle.īecause the three altitudes always intersect at a single point (proof in a later section), the orthocenter can be found by determining the intersection of any two of them. If the triangle is obtuse, the orthocenter will lie outside of it. Notice the second triangle is obtuse, so the altitude will be outside of the triangle. Time to practice Draw an altitude to each triangle from the top vertex. Then draw a second circle using B as center point and the line segment BC as the radius. If the triangle is acute, the orthocenter will lie within it. First draw a circle using A as a center point and the line segment AC as the radius. The location of the orthocenter depends on the type of triangle.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |