Proof of the inscribed circle’s center of an equilateral triangle

 

The inscribed circle’s center of a regular triangle lies at the point of intersection of the medians, heights and perpendicular bisectors of this triangle.

Центр вписанной в равносторонний треугольник окружности

The inscribed circle’s center of a regular triangle

Proof of the property

 

According to the property of the regular triangle, in a regular triangle, the angle bisector drawn to any side is its height, median and perpendicular bisector.

The inscribed circle’s center of a triangle lies at the intersection of the angle bisectors.

Therefore, the inscribed circle’s center of a triangle lies at the intersection of the heights and medians.

Центр вписанной в равносторонний треугольник окружности

Proof of the property