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The proof of Proposition 1.16 given by Euclid is often cited as one place where Euclid gives a flawed proof. [5] [6] [7] Euclid proves the exterior angle theorem by: construct the midpoint E of segment AC, draw the ray BE, construct the point F on ray BE so that E is (also) the midpoint of B and F, draw the segment FC.
Menelaus's theorem, case 1: line DEF passes inside triangle ABC. In Euclidean geometry, Menelaus's theorem, named for Menelaus of Alexandria, is a proposition about triangles in plane geometry. Suppose we have a triangle ABC, and a transversal line that crosses BC, AC, AB at points D, E, F respectively, with D, E, F distinct from A, B, C. A ...
An exterior angle of a triangle is an angle that is a linear pair (and hence supplementary) to an interior angle. The measure of an exterior angle of a triangle is equal to the sum of the measures of the two interior angles that are not adjacent to it; this is the exterior angle theorem. [34]
When θ = π /2, ADB becomes a right triangle, r + s = c, and the original Pythagorean theorem is regained. One proof observes that triangle ABC has the same angles as triangle CAD, but in opposite order. (The two triangles share the angle at vertex A, both contain the angle θ, and so also have the same third angle by the triangle postulate.)
Lie's theorem (Lie algebra) Lie's third theorem ; Lie–Palais theorem (differential geometry) Lindemann–Weierstrass theorem (transcendental number theory) Lie–Kolchin theorem (algebraic groups, representation theory) Liénard's theorem (dynamical systems) Lindelöf's theorem (complex analysis) Lindström's theorem (mathematical logic)
To find an unknown angle, the law of cosines is safer than the law of sines. The reason is that the value of sine for the angle of the triangle does not uniquely determine this angle. For example, if sin β = 0.5, the angle β can equal either 30° or 150°. Using the law of cosines avoids this problem: within the interval from 0° to 180° the ...
The congruence theorems side-angle-side (SAS) and side-side-side (SSS) also hold on a sphere; in addition, if two spherical triangles have an identical angle-angle-angle (AAA) sequence, they are congruent (unlike for plane triangles). [9] The plane-triangle congruence theorem angle-angle-side (AAS) does not hold for spherical triangles. [10]
[4] [5] The triangle inequality is a defining property of norms and measures of distance. This property must be established as a theorem for any function proposed for such purposes for each particular space: for example, spaces such as the real numbers, Euclidean spaces, the L p spaces (p ≥ 1), and inner product spaces.