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Geo.11.21.14- 4.2.notebook

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... Additionally, in triangle AAHB, M is the midpoint of the hypotenuse AB. Thus follows that lines AHM and MB are congruent. Since lines AHM and MB are congruent, angle MBAH and MAHB are congruent, equal to angle value β. Using that fact, BAMAH is equivalent to 180°-2β, and, from this, angle BAMAH is e ...
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... add up to 180◦. But, since the left triangle also has a right angle, its third angle must also be the same size as the third angle above and similarly the third angle of the right triangle is the same as the angle on the left of the intersection point. Thus ...
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Incircle and excircles of a triangle



Incircle redirects here. For incircles of non-triangle polygons, see Tangential quadrilateral or Tangential polygon.In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. The center of the incircle is called the triangle's incenter.An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. Every triangle has three distinct excircles, each tangent to one of the triangle's sides.The center of the incircle, called the incenter, can be found as the intersection of the three internal angle bisectors. The center of an excircle is the intersection of the internal bisector of one angle (at vertex A, for example) and the external bisectors of the other two. The center of this excircle is called the excenter relative to the vertex A, or the excenter of A. Because the internal bisector of an angle is perpendicular to its external bisector, it follows that the center of the incircle together with the three excircle centers form an orthocentric system.Polygons with more than three sides do not all have an incircle tangent to all sides; those that do are called tangential polygons. See also Tangent lines to circles.
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