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If a triangle has a side of length b and an altitude drawn to that side of length h then a similar triangle with corresponding side of length kb will have an altitude drawn to that side of length kh. The area of the first triangle is =, while the area of the similar triangle will be ′ = =. Similar figures which can be decomposed into similar ...
The principle of calculation is that the foot of a cevian is the addition (defined above) of the two vertices (they are the endpoints of the side where the foot lie). For each cevian, the point of concurrency is the sum of the vertex and the foot. Each length ratio may then be calculated from the masses at the points. See Problem One for an ...
The ratio x : y is the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices A and B respectively; the ratio y : z is the ratio of the perpendicular distances from the point to the sidelines opposite vertices B and C respectively; and likewise for z : x and vertices C and A.
Given a cycle of pairs of lobes, the product of the ratios of the volumes of the lobes in each pair is 1. [8] Routh's theorem gives the area of the triangle formed by three cevians in the case that they are not concurrent. Ceva's theorem can be obtained from it by setting the area equal to zero and solving.
In either its simple form or its self-intersecting form, the Lemoine hexagon is interior to the triangle with two vertices on each side of the triangle. [citation needed] Every convex polygon with area can be inscribed in a triangle of area at most equal to . Equality holds only if the polygon is a parallelogram.
The ratio of the area of the incircle to the area of an equilateral triangle, , is larger than that of any non-equilateral triangle. [ 37 ] The ratio of the area to the square of the perimeter of an equilateral triangle, 1 12 3 , {\displaystyle {\frac {1}{12{\sqrt {3}}}},} is larger than that for any other triangle.
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 area of a triangle can be demonstrated, for example by means of the congruence of triangles, as half of the area of a parallelogram that has the same base length and height. A graphic derivation of the formula T = h 2 b {\displaystyle T={\frac {h}{2}}b} that avoids the usual procedure of doubling the area of the triangle and then halving it.