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In each of these cases, the unknown three quantities there are three sides and three angles total can be uniquely determined. Other combinations of sides and angles do not uniquely determine a triangle: Dividing the sides of a triangle in a constant ratio and then drawing lines parallel to the adjacent sides passing through each of these points gives line segments which intersect each other and one of the medians in three places.

Ifthen the extensions of the side parallels intersect the extensions of the medians.

The medians bisect the area of a triangle, as do the side parallels with ratio. The envelope of the lines which bisect the area a triangle forms three hyperbolic arcs. The envelope is somewhat more complicated, however, for lines dividing the area of a triangle into a constant but unequal ratio Dunn and PettyBallWells There are four circles which are tangent to the sides of a triangle, one internal the incircle and the rest external the excircles.

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Their centers are the points of intersection of the angle bisectors of the triangle. Any triangle can be positioned such that its shadow under an orthogonal projection is equilateral. Introduction to Geometry, 2nd ed. Monthly, The Straight Line and Circle.

Riegel und Wiesner, Elements de trigonometrie rectiligne. Charles Babbage Research Foundation, pp. Das Dreieck und seine Beruhungskreise: Cambridge University Press, a. Cambridge University Press, b. Cambridge University Press, c.

Cambridge University Press, d.

Alignments of Remarkable Points of a Triangle. Monthly 72,How to Motivate Teenagers to Do Their Homework.

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