max said:Let sharp triangle ABC inscribed circle $(O;R)$ and $H$ is orthocenter of triangle ABC. circle $(E;r)$ tangent to $HB$, $HC$ and tangent to in circle $(O;R)$.
Prove that: midpoint of $HE$ is center of the circle inscribed the triangle $HBC$
The center of incircle in geometry is the point where the inscribed circle of a triangle or other polygon touches all sides of the shape.
The center of incircle can be calculated using the formula:
(a+b+c)/2 = s
radius = √(s(s-a)(s-b)(s-c))/s
where a, b, and c are the lengths of the sides of the triangle and s is the semi-perimeter.
The center of incircle is important in geometry because it is the largest circle that can be inscribed in a given polygon, and it is tangent to all sides of the shape. It also has many applications in various fields such as architecture, engineering, and mathematics.
The center of incircle is one of the three special centers of a triangle, along with the centroid and circumcenter. It is located inside the triangle, while the centroid is the center of mass and the circumcenter is the center of the circumscribed circle.
No, the center of incircle is always located inside the triangle. However, it can coincide with the centroid or the orthocenter if the triangle is equilateral or right-angled, respectively.