![]() ![]() I think the key part of the definition must be additivity of the measure, as it is in Hilbert's theory. The question is: Can we define angles and sine without referring to Hilbert' theory? Maybe it's possible to define measure of angles in euclidian model R 2. This definition requires proving many theorems (for instance existance of measures and triangle with given angles) and you have to prove that the definition doesn't depnd on the choice of the model, choice of the segments measure and choice of the right triangle. Let x ∈ ( 0, 90 ) and let P be a model of Hilbert's plane euclidian geometry, μ is segments measure, ν is angles measure such that the emasure of the right angle is 90, and △ a b c is a right triangle in which ν ( ∠ a b c ) = x. The radius of the circle is obtained by dropping a perpendicular from the incenter to. It is constructed by taking the intersection of the angle bisectors of the three vertices of the triangle. I know this approach and it's fine, but I'm interested in classical definition. Conversely, if one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle and the angle opposite the diameter is the. In fact, it can be outside the triangle, as in the case of an obtuse triangle, or it can fall at the midpoint of the hypotenuse of a right triangle. The solution may be to define sine and cosine with power series. But I don't know how to define an arc without trigonometric functions. Some people say that the measure of an angle is the ratio of the lenth of the arc to the lenght of the radius. ![]() But there is a problem with defining an angle (and the measure of an angle) without knowing trigonometric functions. I want to define trigonometric function (say sine) formally with the definition that the sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Formal definition of trigonometric functions ∣ O D ‾ ∣ = ∣ O E ‾ ∣ = ∣ O F ‾ ∣ = r, \lvert \overline \rvert. Since the triangle's three sides are all tangents to the inscribed circle, the distances from the circle's center to the three sides are all equal to the circle's radius. In this situation, the circle is called an inscribed circle, and its center is called the inner center, or incenter. A circle is inscribed in the triangle if the triangle's three sides are all tangents to a circle. ![]()
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