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The interior perpendicular bisector of a side of a triangle is the segment, falling entirely on and inside the triangle, of the line that perpendicularly bisects that side. The three perpendicular bisectors of a triangle's three sides intersect at the circumcenter (the center of the circle through the three vertices). Thus any line through a ...
The angle bisector theorem is commonly used when the angle bisectors and side lengths are known. It can be used in a calculation or in a proof. An immediate consequence of the theorem is that the angle bisector of the vertex angle of an isosceles triangle will also bisect the opposite side.
The perpendicular bisectors of any two sides of a triangle intersect in exactly one point. This point must be equidistant from the vertices of the triangle. One way of formulating Thales's theorem is: if the center of a triangle's circumcircle lies on the triangle then the triangle is right, and the center of its circumcircle lies on its ...
As mentioned above, every triangle has a unique circumcircle, a circle passing through all three vertices, whose center is the intersection of the perpendicular bisectors of the triangle's sides. Furthermore, every triangle has a unique Steiner circumellipse, which passes through the triangle's vertices and has its center at the triangle's ...
The perpendicular bisector of a side of a triangle is the line that is perpendicular to that side and passes through its midpoint. The three perpendicular bisectors of a triangle's three sides intersect at the circumcenter (the center of the circle through the three vertices).
A line that is an angle bisector is equidistant from both of its lines when measuring by the perpendicular. At the point where two bisectors intersect, this point is perpendicularly equidistant from the final angle's forming lines (because they are the same distance from this angles opposite edge), and therefore lies on its angle bisector line.
Carnot's theorem: if three perpendiculars on triangle sides intersect in a common point F, then blue area = red area. Carnot's theorem (named after Lazare Carnot) describes a necessary and sufficient condition for three lines that are perpendicular to the (extended) sides of a triangle having a common point of intersection.
The perpendicular bisectors of the sides also play a prominent role in triangle geometry. The Euler line of an isosceles triangle is perpendicular to the triangle's base. The Droz-Farny line theorem concerns a property of two perpendicular lines intersecting at a triangle's orthocenter .