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The three altitudes of a triangle intersect at the orthocenter, which for an acute triangle is inside the triangle. The orthocenter of a triangle, usually denoted by H, is the point where the three (possibly extended) altitudes intersect. [1] [2] The orthocenter lies inside the triangle if and only if the triangle is acute.
Consider a triangle ABC.Let the angle bisector of angle ∠ A intersect side BC at a point D between B and C.The angle bisector theorem states that the ratio of the length of the line segment BD to the length of segment CD is equal to the ratio of the length of side AB to the length of side AC:
The locus of points equidistant from two given points is a straight line that is called the perpendicular bisector of the line segment connecting the points. The perpendicular bisectors of any two sides of a triangle intersect in exactly one point. This point must be equidistant from the vertices of the triangle.
The tangential triangle of a reference triangle (other than a right triangle) is the triangle whose sides are on the tangent lines to the reference triangle's circumcircle at its vertices. [ 64 ] As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the ...
In geometry, symmedians are three particular lines associated with every triangle.They are constructed by taking a median of the triangle (a line connecting a vertex with the midpoint of the opposite side), and reflecting the line over the corresponding angle bisector (the line through the same vertex that divides the angle there in half).
intercept theorem with a pair of intersecting lines intercept theorem with more than two lines. The first two statements remain true if the two rays get replaced by two lines intersecting in , that is, if may be between the parallel lines. [2] The third statement (converse) however does not remain true for lines (see above figure). [3] [4]
Any triangle, in which the altitude equals the geometric mean of the two line segments created by it, is a right triangle. The theorem can also be thought of as a special case of the intersecting chords theorem for a circle, since the converse of Thales' theorem ensures that the hypotenuse of the right angled triangle is the diameter of its ...
A line that is an angle bisector is equidistant from both of its lines when measuring by the perpendicular. At the point where two bisectors intersect, this point is perpendicularly equidistant from the final angle's forming lines (because they are the same distance from this angles opposite edge), and therefore lies on its angle bisector line.