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For acute triangles, the feet of the altitudes all fall on the triangle's sides (not extended). In an obtuse triangle (one with an obtuse angle), the foot of the altitude to the obtuse-angled vertex falls in the interior of the opposite side, but the feet of the altitudes to the acute-angled vertices fall on the opposite extended side, exterior ...
(In a right triangle two of these are merged into the same square, so there are only two distinct inscribed squares.) However, an obtuse triangle has only one inscribed square, one of whose sides coincides with part of the longest side of the triangle. [2]: p. 115 All triangles in which the Euler line is parallel to one side are acute. [3]
The third vertex opposite the base is called the apex. The extended base of a triangle (a particular case of an extended side) is the line that contains the base. When the triangle is obtuse and the base is chosen to be one of the sides adjacent to the obtuse angle, then the altitude dropped perpendicularly from the apex to the base intersects ...
Fig. 5 – An acute triangle with perpendicular. The altitude through vertex C is a segment perpendicular to side c. The distance from the foot of the altitude to vertex A plus the distance from the foot of the altitude to vertex B is equal to the length of side c (see Fig. 5).
For any isosceles triangle, there is a unique square with one side collinear with the base of the triangle and the opposite two corners on its sides. The Calabi triangle is a special isosceles triangle with the property that the other two inscribed squares, with sides collinear with the sides of the triangle, are of the same size as the base ...
One of the oldest and simplest is to take half the product of the length of one side (the base) times the corresponding altitude : [44] =. This formula can be proven by cutting up the triangle and an identical copy into pieces and rearranging the pieces into the shape of a rectangle of base b {\displaystyle b} and height ...
Let D be a point on the line BC, not equal to B or C and such that AD is not an altitude of triangle ABC. Let B 1 be the base (foot) of the altitude in the triangle ABD through B and let C 1 be the base of the altitude in the triangle ACD through C. Then, if D is strictly between B and C, one and only one of B 1 or C 1 lies inside ABC and it ...
For any interior point P, the sum of the lengths of the perpendiculars s + t + u equals the height of the equilateral triangle.. Viviani's theorem, named after Vincenzo Viviani, states that the sum of the shortest distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude. [1]